••'" 


ffiPsi  *""  .-"•••  :v^'-V-:: 


m  m 


SIMPLIFIED  FORMULAS  AND  TABLES 


FOR 


FLOORS,    JOISTS   AND   BEAMS; 
ROOFS,  RAFTERS  AND  PURLINS 


BY 

N.  CLIFFORD  RICKER,  B.S.,  M.ARCH.,  D.AECH. 

Professor  of  Architecture,  University  of  Illinois;  President  Illinois  Board  of  Examiners  of 

Architects;  Chairman  Illinois  Commission  on  Building  Laws;  Honorary  and  Active 

Fellow  American  Institute  of  Architects;    Fellow  American  Association  for 

Advancement   of  Science;    Member   Western  Society   of  Engineers, 

Society  for  Promotion  of  Engineering  Education,  American 

Federation  of  Arts,  Etc. 


FIRST    EDITION 

FIRST    THOUSAND 


NEW  YORK 

JOHN  WILEY  <fe  SONS,  INC. 

LONDON:     CHAPMAN   &  HALL,   LIMITED 

1913 


Copyright,  1913 

BY 
N.  CLIFFORD  RICKER 


THE    SCIENTIFIC    PRESS 

ROBERT    DRUMMOND    AND    COMPANY 

BROOKLYN,    N.    Y. 


PREFACE 


TEXT-BOOKS  on  mechanics  of  engineering  materials 
seldom  make  it  clear,  that  both  formulas  for  safety  against 
rupture  and  for  safety  against  excessive  deflection  must 
be  applied  to  any  structural  problem,  relating  to  members 
supporting  transverse  loads.  Nor  does  any  city  building 
ordinance  known  to  the  author  require  the  use  of  formulas 
to  prevent  excessive  deflection.  Yet  every  competent 
architect  and  engineer  applies  them  in  his  practice,  know- 
ing that  an  excessive  deflection  may  occur  in  long  members, 
entirely  safe  against  rupture,  but  sufficient  to  make  the 
structure  unsightly  and  to  crack  plastering  supported 
by  it. 

The  formulas  for  rupture  and  deflection  usually  given 
are  quite  inconvenient  in  form,  because  large  numbers 
must  be  used  in  computations,  requiring  the  use  of  seven- 
place  logarithms  or  tedious  arithmetical  computations. 

By  transforming  these  formulas,  changing  lengths 
from  inches  to  feet,  loads  from  pounds  to  tons,  constants 
for  the  material  from  pounds  to  tons,  and  bending  moments 
from  inch-pounds  to  foot-tons,  simplifying  the  resulting 
formulas  as  much  as  possible,  they  may  be  put  into  forms 
far  more  convenient  for  practical  use,  and : may  then  be 
grouped  on  a  single  page  for  each  mode  of  support  arid 
arrangement  of  the  loading.  These  simplified  formulas 
can  be  applied  with  sufficient  accuracy  by  using  a  good 
slide  rule  or  a  four-place  table  of  logarithms. 

Tables  have  been  computed  and  are  here  given  for 

the  numerical  values  of     -  and  /  for  rectangular  cross- 

c 


iv  PREFACE 

sections  of  timbers,  and  also  for  the  standard  cross-sections 
of  cast-iron  lintels,  which  make  the  determination  of  their 
proper  sectional  dimensions  as  simple  and  rapid  as  in 
the  case  of  steel  shapes. 

Tables  of  four-place  logarithms  are  also  added  for 
convenience  in  computing. 

This  system  of  formulas  and  tables  has  been  used  for 
several  years  in  my  classes  and  practice,  saving  the  larger 
part  of  the  time  and  labor  usually  required,  and  it  is  now 
published  to  aid  architects  and  engineers  in  their  labors. 

To  extend  the  usefulness  of  the  formulas  and  tables, 
the  proper  method  is  explained  for  applying  them  to 
roofs,  in  order  to  determine  the  safe  dimensions  of  sheath- 
ing, rafters,  and  purlins. 

Finally,  a  series  of  numerical  examples  are  carefully 
worked  to  make  the  proper  use  of  the  work  clearly  apparent. 

N.  CLIFFORD  RICKER. 
URBANA,  ILL.,  March  1,  1913. 


TABLE  OF   CONTENTS 


PAGE 

Preface iii 

ART.    1.  Ordinary  Formulas  for  Beams 1 

ART.    2.  Notation  in  Ordinary  Formulas 1 

ART.    3.  Table  A  of  Ordinary  Formulas 2 

ART.    4.  Inconveniences  in  Use & 

ART.    5.  Method  for  Simplifying  Formulas 4r. 

ART.    6.  Notation  in  Simplified  Formulas 4 

ART.    7.  Method  of  Simplification 5 

ART.    8.  Simplified  Formulas  for  Case  5 & 

ART.    9.  Special  Formulas  for  Any  Material 5 

ART.  10.  Formula  for  Computing  /  from  - 6 

ART.  11.  Formula  for  Safe  Fibre  Stress  and  Deflection 7 

ART.  12.  Actual  Maximum  Deflection 8 

ART.  13.  Formulas  for  Floor  Joists,  Case  5A .8 

ART.  14.  Formulas  for  Flooring,  Case  SB 9 

ART.  15.  General  and  Special  Formulas,  Cases  1  to  10 1O 

ART.  16.  Safe  Values  for  F  and  E 1(> 

ART.  17.  Special  Formulas  for  Common  Materials 11 

ART.  18.  Properties  of  Rectangular  Sections 11 

ART.  19.  Properties  of  Sections  of  Lintels 12 

ART.  20.  Tables  of  Logarithms 14 

ART.  21.  Application  of  Formulas  to  Roofs 14 

ART.  22.  Notation  and  Formulas  for  Roofs 15 

ART.  23.  Sheathing 16 

ART.  24.  Rafters 16 

ART.  25.  Purlins 17 

ART.  26.  Application  of  Formulas  to  Problems 19 

1.  Steel  Girder,  Load  Uniform 19 

2.  Cantilever  Beam,  Load  Uniform 19 

3.  Supported  Beam,  Load  at  Middle 2O 

4.  Steel  Floor  Beam 2O 

5.  Pine  Floor  Joists 2O 

6.  Pine  Floor  Joists 21 

7.  Mill  Deck  Roof 21 

8.  Mill  Roof  Beams 21 

9.  Mill  Roof  Girders.  .  22 


vi  TABLE  OF  CONTENTS 

PAGE 

ART.  27.  Cast-iron  Lintels 22 

10.  Inverted  T-lintel 23 

11.  Box  Lintel  with  Two  Webs 23 

12.  Box  Lintel  with  Three  Webs 23 

13.  Sheathing  of  Roof 24 

14.  Rafters  of  Roof 25 

15.  Purlins  of  Roof 27 

Tables  of  Simplified  Formulas 32 

Case    1.  Beam  Cantilever,  Load  at  Free  End .  . 32,  33 

Case    2.  Beam  Cantilever,  Load  Uniform 34,  35 

Case  2A.  Joist  Cantilever,  Load  Uniform 36,  37 

Case  2s.  Flooring  Cantilever,  Load  Uniform 38,  39 

Case    3.  Beam  Cantilever,  Load  Irregular :  40,  41 

Case    4.  Beam  Supported,  Load  at  Middle 42,  43 

Case    5.  Beam  Supported,  Load  Uniform 44,  45 

Case  5 A.  Joist  Supported,  Load  Uniform . 46,  47 

Case  SB.  Flooring  Supported,  Load  Uniform 48,  49 

Case    6.  Beam  Supported,  Load  Irregular 50,  51 

Case    7.  Beam  Fixed  anc^  Supported,  Load  at  Middle 52,  53 

Case    8.  Beam  Fixed  and  Supported,  Load  Uniform 54,  55 

Case  SA.  Joist  Fixed  and  Supported,  Load  Uniform 56,  57 

Case  SB.  Flooring  Fixed  and  Supported,  Load  Uniform 58,  59 

Case    9.  Beam  Fixed,  Load  at  Middle 60,  61 

Case  10.  Beam  Fixed,  Load  Uniform 62,  63 

Case  10A.  Joist  Fixed,  Load  Uniform 64,  65 

Case  10B.  Flooring  Fixed,  Load  Uniform 66,  67 

Moment  of  Inertia  for  Rectangular  Section 68 

Section  Modulus  for  Rectangular  Section : 69 

Properties  of  Cast-iron  Lintels 70,  71,  72 

Table  of  Logarithms,  0  to  1000 74,  75 

Table  of  Logarithms,  1000  to  2000 76,  77 


SIMPLIFIED  FORMULAS  AND  TABLES 


ERRATA 

Page  1,  second  line  from  bottom.     Read  "  inch-pounds  " 
instead  of  "  inch-tons." 

Page  6,  Art.  10.     Read  "  7  =-  X0.248L^." 

C  £j 

Also   in   fourth   and   second   lines   from   bottom,    read 

F 

-F  instead  of  F. 

iii 

Page  10.  Add  to  the  Table  of  Safe  Values: 
Hemlock.  0.45     (900).  450     (900,000). 

Page  19,  ninth  line  from  top.     Read 
„     30,000X360 
"  8X16,000 

Ninth  line  from  bottom.    Read  "/  =QM7WL2  =0.047  X 
15  X302  =628.5." 

Seventh   line   from   bottom.     Add:     "  by    the    use    of 
simplified  formulas." 

Page  20,  tenth  line  from  bottom.     Read  "/  =  0.047TFL2." 

Page  23,  eleventh  line  from  bottom.     Read  "  and  5^ 
feet  high." 


M' =  maximum  bending  moment  in  inch-tons  acting 
on  the  beam. 


TABLE  OF  CONTENTS 


PAGE 


ART.  27.  Cast-iron  Lintels 22 

10.  Inverted  T-lintel 23 

11.  Box  Lintel  with  Two  Webs 23 

12.  Box  Lintel  with  Three  Webs 23 

13.  Sheathing  of  Roof 24 

14.  Rafters  of  Roof.  . 


SIMPLIFIED  FORMULAS  AND  TABLES 


1.  Ordinary  Formulas  for  Beams. 

The  formulas  for  beams  supporting  transverse  loads, 
commonly  given  in  the  text-books,  are  collected  in  Table  A 
for  comparison  and  reference.  They  evidently  differ 
according  to  the  distribution  of  the  load  along  the  beam, 
and  also  according  to  the  manner  in  which  its  ends  are 
supported  or  fixed.  The  cross-section  of  the  beam  is  here 
assumed  to  be  constant  in  dimensions  and  form  through- 
out its  entire  length,  which  is  always  the  case  for  wooden 
timbers  and  steel  shapes. 

2.  Notation  Employed  in  the  Ordinary  Formulas.    Table  A. 

Let  P  =  total  load  in  pounds  supported  by  the  beam. 
I  =  clear  span  in  inches  of  the  beam. 
S  =  maximum  safe  fibre  stress  in  pounds  per  square 

inch  acting  at  a  cross-section. 
E'  =  modulus  of  elasticity  in  pounds  per  square  inch 

for  its  material. 

E' =  tensile  stress  which  would  theoretically  stretch 
a  bar  1  inch  square  to  twice  its  original  length. 

-  =  section  modulus  of  cross-section  of  beam. 
c 

I  =  section  moment  of  inertia  of  the  same. 

c=  maximum  distance  in  inches  from  horizontal 

gravity  axis  of  cross-section  to  its  most  distant 

fibre. 
A  =  maximum  deflection  of  beam  in  inches,  usually 

limited  to  ^7. . 

oOU 

M '  =  maximum  bending  moment  in  inch-tons  acting 
on  the  beam. 


\  •  MFiHIED  Fi?JliULAS  AND  TABLES 
3.  Table  A.    Formulas  for  Beams. 

«/  PI3 


PI       I  PP 


PP 


4. 


_ 

T~     c 


s-fe 


= 
8         c  384E7* 


P/3 


7. 


/^ 

u 


7P13 

16         c'          7QSET 


10 


12        c         ~  384^7* 


INCONVENIENT  USE  OF  ORDINARY  FORMULAS 


4.  Inconvenient  Use  of  Ordinary  Formulas. 

As  an  illustration,  take  the  following  practical  example. 
A  steel  beam  is  to  be  composed  of  two  steel  I-beams, 
is  30  ft.  long  and  must  safely  support  a  uniformly  dis- 
tributed load  of  20,000  Ibs.  Its  most  economical  cross- 
section  and  actual  maximum  deflection  are  to  be  determined. 
For  rolled  steel  shapes,  S  =  16,000  Ibs.,  E'  =29,000,000, 
and  1=  360  ins. 

By  formulas  for  Case  5,  Table  A: 

PT     QI  I      PI      20000x360 

-g-  =  S  -;    transposing;    -  =  ^  -     8xl6000   !-  56.12. 

5  P  I3 
A  =  ,    ;    transposing  ; 


75  P  I2      75  X20000  x  129600 
16  #'  16X29000000 

Since  two  I-beams  are  to  be  used,  for  each,  -  =  28.06  and 
7  =  209.69. 

By  "Cambria":  two  12  in.,  31.5  Ib.  I-beams  will  suffice 
for  both  values. 

For  the  selected  section,  1=  2x215.8  =  431.6  for  both 
beams. 

5PZ3          5X20000X46656000 
~  384  #'/  "  384X29000000X431.6  ~ 

Since    this    maximum    deflection    should    not    exceed 
1  in.,  this  compound  beam  may  be  safely  employed. 


Even  with  the  use  of  logarithms  in  solving  this  problem, 
it  is  evident  that  the  use  of  the  ordinary  formulas  requires 
considerable  time  and  a  large  number  of  figures,  with 
possible  errors  in  the  computations,  and  that  they  are 


4  SIMPLIFIED  FORMULAS  AND  TABLES 

not  adapted  for  use  on  the  slide  rule.  Also,  that  if  these 
formulas  can  be  materially  simplified,  much  time  and 
labor  can  be  saved,  and  it  may  become  entirely  possible 
to  make  the  necessary  calculations  with  four-place  loga- 
rithms or  a  good  slide  rule,  obtaining  results  sufficiently 
accurate  for  any  practical  purpose. 


5.  Method  for  Simplifying  the  Ordinary  Formulas. 

The  following  changes  are  made  in  the  ordinary  formulas 
in  Table  A: 

a.  Change  load  on  beam  from  pounds  to  tons. 

b.  Change  numerical  values  of  S  and  E'  in  pounds  to  F 
and  E  in  tons. 

c.  Change  length  of  beam  from  I  in  inches  to  L  in  feet. 

d.  Change  bending  moments  from  inch-pounds  to  foot- 
tons. 

Other  values  remain  as  before. 


6.  Notation  Employed  in  the  Simplified  Formulas. 

Let  W  =  total  load  on  beam  in  tons. 

.  w  =  total  load  in  pounds  per  square  foot  of  a  floor. 
L=  length  of  beam  in  feet,  or   distance  between 

centres  of  beams. 

e  =  distance  in  inches  between  centres  of  floor  joists. 
t=  thickness  in  inches  of  the  flooring. 
F=  maximum  safe  fibre  stress  in  tons  per  square 

inch. 

E=  modulus  of  elasticity  in  tons. 
M  =  maximum  bending  moment  in  foot-tons. 
A  =  maximum  deflection  of  beam  in  inches;    should 

,  L 
not  exceed  o?j. 


METHOD  OF  SIMPLIFICATION 


7.  Method  of  Simplification. 

In  simplifying  or  transforming  a  formula,  care  must 
always  be  taken  to  preserve  the  numerical  value  of  each 
side  of  the  equation  representing  the  formula. 

As  an  example  of  the  application  of  the  method,  take 
the  ordinary  formulas  given  for  Case  5  in  Table  A. 

Substitute  2000  W  for  P;     12  L  for  Z;    2000  P  for  S: 

2000  #  for  E']    and  ™  for  A.     Then  reduce  the  equation 
oU 

to  its  simplest  form  and  transpose  to  obtain  the  forms 
most  convenient  for  use. 

PI  .aJ.2000TTxl2L  J  p7 

8  c  8  c  c 

5PZ3       5X2QQOTFX1728L3       L       22.5 
A~ 


384#'I~        384X2000^7        "  30  El 


8.  General  Simplified  Formulas  for  Case  5. 

The  formulas  just  obtained  may  be  put  into  forms  more 
convenient  for  use. 

15WL-F1- 


c  30  "        El 

I      1.5TFL  675  TFL2      section  mom- 

-  = ^ —  =  section  modulus.  /  = ^ =          , . 

c  F  E  ent  of  inertia. 

=  safe  load. 
=  safe  length. 


SIMPLIFIED  FOEMULAS  AND  TABLES 


9.  Special  Formulas  for  any  Material. 

For  example,  take  the  general  formulas  just  found, 
adapt  them  to  steel  by  substituting  the  numerical  values 
for  F  and  E  and  reduce  to  simplest  form. 


5.333  145QQ  / 


/ 
~  \ 


10.  Formula  for  Directly  Computing  the  Numerical  Value  of  I 

from  that  of  — . 

Evidently  for  a  beam  of  a  given  length,  load,  and 
material,  the  numerical  values  in  the  preceding  general 
formulas  for  W  are  equal,  may  be  equated  and  simplified 
for  7. 

Then 

T          7?  T?T  T 

~  X 1  K  j  =  anr.  r,  from  which  is  found  I  =  -  X450  L  F. 
c     1.5L      675  L'  c 

Therefore  in  Case  5,  after  obtaining  the  numerical 
value  of  — ,  it  may  save  time  to  multiply  this  value  by 

0 

450  L  F  instead  of  using  the  formula  given  in  Art.  9  for  7. 
This  formula  may  also  be  simplified  by  inserting  the  value 
of  F  and  reducing,  making  it  very  convenient  for  the  slide 
rule. 


FORMULA  FOR  MAXIMUM  SAFE  FIBRE  STRESS 


11.  Formula  for  Maximum  Safe  Fibre  Stress  and  Deflection. 

The  preceding  formulas  for  safety  against  rupture 
and  excessive  deflection  are  entirely  independent  of  each 
other.  Therefore,  if  a  beam  of  any  given  material  and 
uniform  cross-section  be  assumed,  its  safe  load  W  be 
computed  by  both  formulas  for  successive  lengths  L, 
and  the  values  of  W  be  plotted,  two  curves  will  be  obtained 
and  intersect  at  a  common  point  at  which  the  numerical 
values  of  W  and  L  will  be  respectively  equal,  as  illustrated 
in  Fig.  11.  Hence  for  the  intersection,  we  may  equate  the 


12r 

10 


10  15  20 

L.   IN   FEET. 

FIG.  11. 


values  of  W  in  the  two  equations,  obtaining  in  Case  5, 
EC 


L  = 


450  F' 


For  F  and   E  may  then   be   substituted  the 


numerical  values  for  any  material,  thus  producing  a  very 
simple  formula,  so  that  L  can  be  found  by  it  directly. 

For  lengths  less  than  L  by  this  formula,  the  formula 
for  safety  against  rupture  gives  safest  results;  for  those 
greater  than  L,  the  formula  against  excessive  deflection  is 
safest.  Hence  if  this  value  of  L  be  known  for  any  material, 
it  is  only  necessary  to  apply  one  formula  below  it  and  the 
other  above  it. 


SIMPLIFIED  FORMULAS  AND  TABLES 


12.  Actual  Maximum  Deflection. 

This  formula  gives  the  actual  maximum  deflection  of 
the  beam  in  inches. 

5  PI3    _  5X2000TTX1728L3  _  22.5  WL3 
A~384#'7~       384X2000  ~ 


13.  General  Formulas  for  Floor  Joists.    Case  5  a. 

Let  e  =  distance  in  inches  between  centres  of  joists. 

w=  total  live  and  dead  loads  in  pounds  per  square 
foot  of  floor. 

e_       w     _       _  wL  e 

T2X2000~       "24000* 

Substituting  this  value  for  W  in  the  general  formulas 
for  W  and  simplifying, 

/      1.5  WL      1.5wL2e        wL2e 


c  F        "  24000^    "  16000  F' 

675  WL2     675  w  L3e       wL3e 
E        ''  24000  #  "  35.56^' 

Then  by  transposition: 

I       w  L2e  w  L3e 


c     16000  F'  35.56 

I     16000  F  35.56 


7     16000  F  35.56  E  I 

cX    wL2    '  e=       wL3    ' 


/7 
~  \  c 


16000  F  L 

c        we'  \      w  e 


GENERAL  FORMULAS  FOR  FLOORING  9 

By  inserting  the  values  of  F  and  E,  these  general  for- 
mulas are  changed  into  simpler  formulas  for  any  particular 
material. 

The  formulas  for  directly  computing  7  from  -  and  for  L 

c 

for  maximum  safe  fibre  stress  and  deflection  are  unchanged 
from  those  found  for  Case  5. 

For  actual  deflection  of  a  joist,  substituting  value  of  W 
and  simplifying: 

_  22.5  wL*e         w  L*e 
A  ~  24000  El"  1067  E  F 


14.  General  Formulas  for  Flooring.     Case  5  b. 

Let  t=  thickness  in  inches  of  the  flooring. 
Take  e  =  12  ins.,  assuming  a  strip  of  floor  1  ft.  wide. 
Then  for  the  rectangular  section  of  a  floor  board: 


~c=~S~-      6  =  12=^2~: 

Substituting  12  for  e,  2  t2  for  -,  and  t3  for  /  in  equations 

c 

for  floor  joists  and  simplifying: 

12j^  12  wL* 

*l  ~~  16000  F'  ~  35.56  #' 


w  L2  w  . 


.96  # 
2667  F  t2  2.96  E  t3 


These  formulas  may  be  further  simplified  by  inserting 
the  values  of  F  and  E  for  the  particular  material. 


10 


SIMPLIFIED  FORMULAS  AND  TABLES 


The  general  formula  for  maximum  safe  fibre  stress  and 
deflection  is  obtained  by  equating  the  values  just  found 
for  w  and  simplifying. 

Et 


L  = 


933 


The  general  formula  for  actual  deflection  is  obtained 
by  substituting  t3  for  /  in  the  formula  for  actual  deflection 
of  a  joist  and  reducing. 

w  L4e 
A  =  1067  E  t3' 

15.  General  and  Special  Formulas  for  Cases  1  to  10. 

These  are  derived  from  the  ordinary  formulas  given 
in  Table  A  by  the  method  just  explained  and  applied  to 
those  of  Cases  5,  5  a  and  5  b. 


16.  Numerical  Safe  Values  recommended  for  F  and  E. 


Material. 

F.          Lbs. 

E.               Lbs. 

Cedar  

0.45       (900) 

450       (900,000) 

Cypress  

0.50    (1,000) 

550    (1,100,000) 

Fir,  Washington  .  .  . 

0.70    (1,400) 

700    (1,400,000) 

Gum  

0.55    (1,100) 

650    (1,300,000) 

Iron,  cast.    Tension 

1.50    (3,000) 

8,000  (16,000,000) 

Iron,  wrought  

6.00  (12,000) 

14,000  (28,000,000) 

Maple,  sugar  

0.75    (1,500) 

800    (1,600,000) 

Oak,  white  

0.65    (1,300) 

750    (1,500,000) 

Pine,  longleaf  

0.70    (1,400) 

850    (1,700,000) 

Pine,  Norway  

0.50    (1,000) 

600    (1,200,000) 

Pine,  pitch  

0.55    (1,100) 

600    (1,200,000) 

Pine,  shortleaf  

0.55    (1,100) 

600    (1,200,000) 

Pine,  white  

0.45      (900) 

500    (1,000,000) 

Poplar,  yellow  

0.45       (900) 

500    (1,000,000) 

Redwood  

0.40       (800) 

350       (700,000) 

Spruce  

0.55    (1,100) 

650    (1,300,000) 

Steel  shapes  

8.00  (16,000) 

14,500  (29,000,000) 

FORMULAS  FOR  THE  COMMONLY  USED  MATERIALS  11 

These  are  safe  average  values,  based  on  the  results  of 
experiments  and  the  average  requirements  of  the  building 
ordinances  of  the  principal  cities  in  the  United  States. 
The  corresponding  safe  values  for  any  other  materials,  or 
those  prescribed  by  any  building  ordinance,  may  easily 
be  inserted  in  the  general  formulas  for  the  particular  case, 
then  simplified  to  obtain  the  working  formulas. 

17.    Special  Formulas  for  the  Commonly  Used  Materials. 

From  the  simplified  general  formulas  for  Cases  1  to  10, 
by  substituting  the  proper  values  of  F  and  E  taken  from 
Art.  16  and  simplifying,  are  derived  the  special  formulas 
here  given  for  steel,  cast  iron,  Washington  fir,  hemlock, 
white  oak,  longleaf,  shortleaf,  and  white  pine,  and  for 
spruce.  These  materials  have  been  selected  because  they 
are  more  commonly  employed  in  the  Middle  and  Eastern 
States.  These  special  formulas  are  then  most  rapidly 
applied  by  using  four-place  logarithms  or  a  good  slide  rule. 

18.  Tables  of  Properties  of  Rectangular  Sections. 

Tables  19  and  20  are  to  be  used  in  determining  the 

dimensions  of  timbers  corresponding  to  the  values  of 

c 

and  I  obtained  by  the  formulas.  The  upper  horizontal 
line  of  figures  represents  the  horizontal  breadth  of  the 
section,  and  the  left-hand  vertical  line  contains  the  vertical 

depth.     The   numerical   values  of  -  in  Table  19  are  com- 

c 

puted  by  the  usual  formula, 

I    bd2 
c~   6  * 

Those  of  /  in  Table  20  are  obtained  by  the  formula, 

bd* 
2~  12' 


12 


SIMPLIFIED  FORMULAS  AND  TABLES 


19.  Tables  of  Properties  of  Sections  of  Cast-iron  Lintels. 

These  tables  include  the  stock  sections  and  sectional 
dimensions  of  lintels  usually  furnished  by  the  large 
foundries.  It  is  not  economical  to 
design  other  sections,  excepting  when 
a  considerable  number  are  to  be  cast 
from  the  new  pattern  required.  Lin- 
tels are  now  generally  composed  of 
pairs  of  steel  I-beams.  Cast-iron 
lintels  should  only  be  used  in  Case 
4,  5,  or  6,  since  their  design  becomes 
too  complex  in  the  other  cases. 

Fig.  12  is  the  section  of  an  in- 
verted T-section,  also  applicable  to  an  L-section;  Fig.  13 
is  that  of  a  box  lintel;  and  Fig.  14  is  a  box  lintel  with 


ii 

r- 

! 
1 

t 

t-- 

-M 

4 

_l__t_ 

7 

F 

[Q. 

12. 

</> 

r-f 
i 
i 
i 
i 

! 

f~T~E 

^  i 

4 

--Jr—  !--^- 

o 

FIG.  13. 

']' 


k 


FIG.  16. 


three  webs.  Flanges  and  webs  have  equal  thickness  of 
metal,  and  they  are  to  be  connected  at  proper  distances 
by  cross  webs  to  prevent  crippling. 


PROPERTIES  OF  SECTIONS  OF  CAST-IRON  LINTELS     13 

The    formulas    employed    in    the    computations    were 
obtained  as  follows: 

Let  t  =  uniform  thickness  of  metal  in  inches. 
h  =  height  of  webs  from  flange  in  inches. 
b  =  breadth  of  flange  in  inches. 
A  =  total  sectional  area  of  lintel  in  square  inches. 
A'  =  total  sectional  area  of  webs  in  square  inches. 
A"  =  total  sectional  area  of  flange  in  square  inches. 
d=  vertical    distance    in    inches    between  horizontal 

gravity  axes  of  webs  and  flange. 
df  =  vertical  distance  in  inches  between  gravity  axis 

of  webs  and  neutral  axis  of  entire  section. 
d,,    =  vertical  distance  in  inches  between  gravity  axis 

of  flange  and  neutral  axis  of  entire  section. 
c  =  distance  in  inches  between  bottom  of  flange  and 

neutral  axis  of  section. 
7=  moment  of  inertia  of  the  entire  section  about  its 

neutral  axis. 

/'=  moment  of  inertia  of  all  webs  about  their  hori- 
zontal gravity  axis. 

/"=  moment  of  inertia  of  flange  about  its  horizontal 
gravity  axis. 

-  =  section  modulus  of  entire  section  about  its  neutral 
c 

axis  on  tension  side. 
Then       d  =—^—  =  half  depth  of  lintel  in  inches. 

Also   for   location   of   the   neutral   axis   of   the   entire 
section. 

A'd 

A  :  Af  :  d  :  dlt\     hence  du  =  — r- . 

A. 

By  the  usual  formula  for  /  about  any  axis  parallel  to 
its  gravity  axis: 

I=r+AfdJ2+rf+A"d,,2=  moment    of    inertia    of    entire 
section. 

Also       c  =  dn+-^,     and    -=  section  modulus. 

—  c 


14  SIMPLIFIED  FORMULAS  AND  TABLES 


20.  Tables  of  Logarithms. 

In  order  to  make  this  work  as  convenient  as  possible, 
two  tables  of  four-place  logarithms  have  been  added  in 
Tables  24  and  25,  one  extending  from  0  to  999,  the  other 
from  1000  to  1999.  These  will  be  found  sufficient  for 
solving  problems  relating  to  beams,  joists,  and  flooring. 
Or  a  good  slide  rule  may  be  employed,  saving  some  time 
and  the  labor  of  writing  down  the  logarithms,  but  with 
more  liability  to  error  in  locating  the  decimal  point. 


21.    Application  of  Formulas  and  Tables  to  Roofs. 

These  simplified  formulas  may  be  applied  to  roofs  as 
well  as  to  floors,  in  the  following  manner. 

Loads  on  roofs  are  composed  of  four  different  kinds: 

1.  Permanent    loads    in    pounds    per    square    foot    of 
inclined  surface,  acting  vertically,  and  consisting  of  weight 
of  covering,  sheathing,  rafters,  and  purlins. 

2.  Snow  load   in   pounds   per   horizontal   square   foot, 
acting  vertically,  its  magnitude  varying  from  0  to  35  Ibs., 
according  to  latitude. 

3.  Wind  load  or  pressure  in  pounds  per  square  foot  of 
inclined  surface,  acting  at  right  angles  to  the  latter,  its 
magnitude  varying  from  0  to  50  Ibs.,  according  to  exposure 
and  inclination  of  the  roof. 

4.  Accidental  loads,   for  example,   25  Ibs.   per  square 
foot  of  a  flat  roof  for  weight  of  snow,  firemen,  etc.     Acts 
vertically. 

The  weight  of  the  trusses  supporting  the  roof  is  not 
included  here. 


FORMULAS  EMPLOYED  FOR  LOADS  ON  ROOFS    15 

22.  Notation  and  Formulas  Employed  for  Loads  on  Roofs. 

Let  p  =  permanent  load  in  pounds  per  square  foot  of 

inclined  surface, 
s  =  snow  load  in  pounds  per  square  foot  of  horizontal 

surface. 
w  =wind  load  in  pounds  per  square  foot  of  inclined 

surface. 
i°  =  angle  of  inclination  of  surface  from  horizontal. 

Then  s  cos  i  =  snow  load  in  pounds  per  square  foot  of 
inclined  surface. 

For  a  flat  roof,  cosi  =  l,  w=Q'}  the  roof  is  then  treated 
like  a  floor. 

p  cos  i°  =  normal  component  of  permanent  load  p. 

p  sin  i°  =  parallel  component  of  permanent  load  p. 

s  cos2  i°          =  normal  component  of  snow  load  s  cos  i. 
s  sin  i°  cos  i°  =  parallel  component  of  snow  load  s  cos  i. 
w  =  normal  component  of  wind  load  w. 

0  =  parallel  component  of  wind  load  w. 

Since  the  maximum  snow  load  and  wind  load  can 
scarcely  occur  simultaneously  on  the  roof  surface,  we 
may  have  either  one  of  two  cases. 

a.  Permanent  and  snow  loads  form  the  maximum  load- 
ing. 

cos  i  (p+s  cos  i)  =  normal  component  of  p  and  s  loads, 
sin  i(p+s  cos  i)  =  parallel  component  of  p  and  s  loads. 

6.  Permanent  and  wind  loads  form  the  maximum 
loading. 

p  cos  i+w  =  normal  component  of  p  and  w  loads. 
p  sin  {+0  =  parallel  component  of  p  and  w  loads. 

Either  pair,  a  or  6,  of  formulas  are  to  be  employed, 
which  corresponds  to  the  mode  of  loading,  that  produces 
the  maximum  stresses  in  the  roof. 


16  SIMPLIFIED  FORMULAS  AND  TABLES 

23.  Sheathing. 

Here  p=  weight  of  covering -f  weight  of  sheathing  per 
inclined  square  foot. 

For  an  inclined  roof  the  parallel  component  of  this 
loading  may  usually  be  neglected,  since  it  is  safely  resisted 
by  the  edgewise  strength  of  the  sheathing.  Take  the 
maximum  normal  component,  substitute  this  for  w  in  the 
formulas  of  Case  5  b  to  determine  L  =  maximum  safe  distance 
in  feet  between  centres  of  the  supporting  rafters. 


24.  Rafters. 

Here  p=  weight  of  covering  +  weight  of  shea  thing  + 
average  weight  of  rafters  per  inclined  square  foot. 

The  maximum  normal  component  acts  transversely  and 
its  value  is  substituted  for  w  in  the  formulas  of  Case  5  a 

to  determine  -  and  7;    the  dimensions  of  cross-section  of 
c 

rafters  are  then  found.     By  applying  the  formula  for  A, 
Case  5  a,  the  maximum  deflection  A  of  the  rafter  is  found. 
The  parallel  component  of  the  loading  acts  lengthwise 
the  rafter  producing  compression.     The  magnitude  of  this 

, .      ,.      ^         e  L  X  par.  component 
compression  at  mid-length  of  ratter  =  -  4&QQQ 

in  tons. 

Let  u=  uniform  compression  in  tons  per  square  inch  at 

this  section  of  rafter. 

d  =  depth  of  rafter  in  inches,  for  rectangular,  I  or 
channel  section. 

((\  A\ 
1  +-r)  =  maximum  compression  in  top  fibres  in 

tons  per  square  inch. 

This  is  then  to  be  deducted  from  the  value  of  F 
employed  for  the  material  in  the  formulas  of  Case  5  a; 


PUELINS  17 

substitute  the  remainder  for  F  in  the  general  formula  and 

compute  anew  the  proper  values  of  -,  7  and   dimensions 

c 

of  rafter.     In  all  roofs  of  ordinary  inclination,  this  parallel 
component  may  be  neglected. 


25.  Purlins. 

Here  p=  weights   of   covering + sheathing + average  for 
rafters  +  average  for  purlins  per  inclined  square  foot. 
Purlins  may  be  set  in  either  of  three  ways : 

a.  With  middle  or  major  axial  plane  containing  resultant 
of  all  loads  on  purlin.     But  these  loads  are  liable  to  varia- 
tion,   and   this  resultant   then  varies  in  magnitude   and 
direction. 

b.  Major  axial  plane  at  right  angles  (normal)  to  roof 
surface. 

Let  W=  total  load  in  tons   on  purlin  uniformly   dis- 
tributed. 

W  =  normal  component  of  loads  on  purlin. 
W"  =  parallel  component  of  loads  on  purlin. 

c.  Major  axial  plan  vertical  and  making  angle  j°  with 
resultant  of  maximum  simultaneous  loads  on  purlin. 

W  =  W  cos  j °  =  vertical  component  of  loads  on  purlin. 
W"  =  W  sin  j°  =  horizontal  component  of  loads  on  purlin. 

After  obtaining  the  component  W,  which  acts  in  the 
major  axial  plane  of  the  purlin,  and  W",  that  acts  at  right 
angles  to  the  former,  the  formulas  of  Case  5  are  applied 

to  obtain  -  and  of  I  for  each  component.     A  section  is  then 
c 

selected  that  has  the  required  values  of  -  and  I  in  the  two 

c 

directions. 


18  SIMPLIFIED  FORMULAS  AND  TABLES 

1.  For  a  timber  purlin,  the  required  sectional  dimen- 
sions may  be  found  by  Tables  19  and  20,  selecting  a  section 

possessing  the  required  values  of  —  and  I  in  the  respective 

c 

directions. 

2.  For  a  steel  purlin,  which  may  be  composed  of  two 
I-beams  latticed  together  and  spaced  apart  sufficiently  to 
have  the  required  stiffness  sidewise.     Or,  more  commonly, 

a  single  I-beam  is  used  with  the  required  values  of  -  and  / 

G 

for  the  component  W '.  This  beam  is  then  subdivided  in 
equal  spans  by  one  or  more  suspension  rods  extending  up 
to  the  ridge  of  the  roof,  so  that  its  stiffness  sidewise  is 
sufficient  for  each  short  span. 

But  since  the  neutral  axis  of  the  purlin  is  not  usually 
at  right  angles  to  its  major  axial  plane,  the  angles  of  this 
section  will  not  be  equidistant  from  this  neutral  axis,  and 
those  more  distant  will  be  more  stressed,  than  if  the  neutral 
axis  were  parallel  to  the  top  of  the  purlin. 

Therefore,  the  following  formula  is  then  to  be  applied 
to  determine  the  maximum  fibre  stress  found  in  these  more 
distant  angles,  and  whether  it  exceeds  the  safe  limit  for  the 
material  used. 

Let  b  =  parallel  breadth  of  the  purlin  in  inches. 
d  =  normal  depth  of  purlin  in  inches. 
Iv=  moment  of  inertia  about  parallel  minor  axis  of 

section. 
Ix=  moment  of  inertia  about  normal  major  axis  of 

section. 

/W'd     W"b\ 

Then  0.75  L{  -j — I — j — )  =  maximum  fibre  stress  in  tons 
\  lv        !•«,•/ 

per  square  inch. 

If  this  exceeds  the  safe  value  for  the  material,  a  larger 
section  must  be  taken,  until  a  sufficient  one  is  obtained. 
This  formula  must  be  applied  to  purlins  of  wood  or  steel 
excepting  when  W  coincides  with  the  major  axial  plane  of 
the  cross-section. 


APPLICATION  OF  FORMULAS  TO  PROBLEMS  19 


26.  Application  of  Formulas  to  Problems. 

Some  problems  will  illustrate  the  practical  use  of  the 
formulas  and  tables. 

PROBLEM  1.  A  steel  girder  is  30  ft.  long  and  must 
safely  support  a  uniform  load  of  15  tons.  To  be  composed 
of  two  I-beams  with  separators  and  bolts. 

a.  By  ordinary  formulas,  Case  5,  Table  A. 

p  7        a  r 

For  safety  against  rupture:  ~Q-  =  —  • 

o  C 

I     PI     30000+360 
Transposing  :  -  =        =  =  84'38' 


5P  I3 

For  safety  against  excessive  deflection  :  A  = 


Transposing  : 

_  5X360PZ2  _  5X360X30000X129600 
384  E  384x29000000 

b.  By  simplified  formulas,  Case  5,  Table  7. 
For  safety  against  rupture: 

-  =  0.187  W  L  =  0.187  X  15  X30  =  84.4. 
c 

For  safety  against  deflection: 

/=  1.192TFL2  =  1.192  X  15  X302=  628.5 

By  Cambria,  2,  15  in.,  42-  Ib.  I-beams  are  required. 
Comparison  shows  a  decided  economy  in  time  and  labor  in 
computations. 

PROBLEM  2.  Beam  cantilever  with  uniform  load, 
Case  2,  Table  2.  Free  length  10  ft.,  and  supporting  a 
load  of  0.5  ton  per  foot.  Washington  fir. 

-  =  8.58  W  L  =  8.58  X  5  X  10  =  429. 

C/ 

/  =  9.26  W  L2  =  9.26  X5  X  102  =  4630. 


20  SIMPLIFIED  FORMULAS  AND  TABLES 

By  Table  19  for-:  8x18,  10x16,  12x16,  14x14  ins. 

O 

By  Table  20  for  I:  8x20,  10x18,  12x18,  14x16  ins. 
Therefore  the  beam  may  be  made  10x18  or  14x16,  as 
most  convenient. 

PROBLEM  3.  Beam  supported  at  ends  with  load  at 
middle.  Case  4,  Table  6.  Beam  of  shortleaf  pine  16  ft. 
clear  span,  which  must  safely  support  a  load  of  3  tons  at 
middle  of  span. 

-  =  5.45  W  L  =  5.45  X3  X16  =  262. 
c 

1=  1.802  WL2=  1.802  X3X162=  1384. 

By  Table  19  for  -:  4x20,  6x18,  8x14,  10x14,  12x12. 
c 

By  Table  20  for  7:  4x18,  6x16,  8x14,  10x12,  12x12. 
Most  economical  to  make  the  section  8x14  ins. 

PROBLEM  4.  Steel  floor  beam  supporting  hollow  tile 
floor,  Case  5,  Table  7.  Beam  16  ft.  long  and  set  4  ft.  on 
centres.  Must  safely  support  a  total  live  and  dead  load 
of  146  Ibs.  per  square  foot  of  floor. 

146 
Here  W  =          X4  X  16  =  4.673  tons. 


-  =  0.187  WL  =0.187x4.673x16  =13.98. 
c 

I  =  0.046  W  L2  =  0.047  X  4.673  X  162  =  56.22. 

By  Cambria:  1,  8  in.,  18  Ib.  I-beam  just  suffices. 

PROBLEM  5.  Joists  supporting  floor  and  ceiling,  Case  5  a, 
Table  8.  Shortleaf  pine  joists  18  ft.  long  and  set  16  ins.  on 
centres  must  safely  support  a  total  live  and  dead  load  of 
65  Ibs.  per  square  foot. 

I     wL2e65xl82Xl6 


_ 
8800  8800 


_       0 


_  wL*e  _  65  X  183  Xl6 
1  ~  21337  ~       21337        ~  ^  ' 


APPLICATION  OF  FORMULAS  TO  PROBLEMS  21 

By  Table  19  for-:  If  Xl2,  2x12,  3x10,  4x8. 
c 

By  Table  20  for  I :  If  X 14,  2  X 12,  3  X 12,  4  X 10. 
Hence  the  joists  should  either  be  If  Xl4  or  2  Xl2,  full  size. 

PROBLEM  6.  Joists  for  schoolroom  floor,  Case  5  a, 
Table  8.  Joists  of  longleaf  pine,  24  ft.  long,  set  12  ins.  on 
centres,  safely  supporting  total  live  and  dead  load  of  102  Ibs. 
per  square  foot  of  floor. 

I  _  wL2e  _  1Q2X242X12 

c~  11200"         11200          "Tv? 


T_  _  102x243Xl2 

29606"        29606        ~ 


By  Table  19  for-:  3x12,4x10. 

C/ 

By  Table  20  for  /:  3  X 14,  4x12. 
Therefore  it  is  best  to  make  the  joists  3  Xl4  ins. 

PROBLEM  7.  Mill  construction  for  deck  roof,  Case  5  6, 
Table  9.  Plank  roof  of  2f  ins.  shortleaf  pine,  which  must 
safely  support  a  total  live  and  dead  load  of  40  Ibs.  per 
square  foot.  First  find  maximum  safe  distance  between 
centres  of  supporting  beams. 


38.3 1     38.3X2.625  _Q 

L  =  — 7=-=  -    -7= —  =•  15.89  ft.  on  centres. 
Vw  v40 


T     I2.lt     12.1X2.625 

L  =    3 —  =  -      3/-^    ~  =  9.29  ft.  on  centres. 
V40 


Therefore,  the  supporting  beams  cannot  be  safely  set 
over  9.4  ft.  on  centres. 

PROBLEM  8.  Mill  roof  beams,  Case  5,  Table  7.  Assum- 
ing the  roof  beams  to  be  set  8  ft.  on  centres  and  to  be  of 
shortleaf  pine  also,  and  16  ft.  in  clear  length. 


22  SIMPLIFIED  FOEMULAS  AND  TABLES 

W  =8X16X42=  5376  Ibs.  =  2.688  tons,  allowing  2  Ibs. 
per  square  foot  for  average  weight  of  roof  beams. 

-=2.730  WL  =2.730X2.688X16  =117.4. 

C/ 

/  =  1.125  W  L2  =  1.125  X2.688  X 162  =  774.4. 

By  Table  19 :  4  X 14,  6  X 12,  8  X 10. 

By  Table  20 :  4  X 14,  6  X 12,  8  X 12. 

Therefore  6  Xl2  beams  are  preferable. 

PROBLEM  9.  Mill  roof  girders,  Case  4,  Table  6.  Assum- 
ing that  the  posts  are  16  ft.  on  centres,  that  one  inter- 
mediate beam  is  supported  at  middle  of  girder,  for  shortleaf 
pine  girders. 

-  =  5.45  W  L  =  5.45  X2.688  X 16  =  234.4. 
c 

1=  1.802  WL2=  1.802  X2.688X162=  1240.3. 

By  Table  19:  6x16,8x14,10x12. 
By  Table  20:  6x14,  8x14,  10x12. 
Hence  it  will  be  best  to  make  these  girders  8  X 14  ins. 


27.  Cast-iron  Lintels. 

Although  lintels  composed  of  steel  shapes  are  now 
generally  employed  to  span  openings  in  masonry  walls, 
cast-iron  lintels  are  still  frequently  used  for  this  purpose. 
But  only  certain  stock  sections  and  sizes  are  usually  fur- 
nished by  the  larger  foundries,  since  a  specially  made 
pattern  would  usually  make  the  cost  of  a  few  lintels 
prohibitive.  Tables  21,  22,  and  23  comprise  the  standard 
forms  and  dimensions  of  lintels  usually  furnished.  For 
these  have  been  carefully  computed  their  properties, 


CAST-IRON  LINTELS  23 

i.e.,   the  numerical    values    of    -,  /,  and  c  =  distance  in 

c 

inches  from  bottom  of  lintel  section  to  its  horizontal 
gravity  axis.  Thus,  it  now  becomes  possible  to  apply  the 
formulas  previously  given  to  determine  the  required  cross- 
section  of  a  cast-iron  lintel  as  easily  as  to  obtain  the 
dimensions  of  a  beam  of  wood  or  of  steel  shapes. 

PROBLEM  10.  An  inverted  T-lintel  is  16  ft.  long  with  a 
section  8x12  ins.  and  1J  in.  metal.  Determine  its  safe 
uniform  load  W. 

By  Table  22:  -  =  58.2;    /  =  120.8.     Case  5,  Table  7. 
c 

W=-  X^  =  58.2  X^  =  3.637  tons. 
c       L  16 

'  J  i  Of)  Q 

W  =  11.85     ,  =  11.85  X-        =  5.592  tons. 


Therefore,  the  maximum  safe  uniform  load  of  the  lintel  = 
3.637  tons. 

PROBLEM  11.  Box  lintel,  with  two  webs  and  uniformly 
loaded.  Clear  span  of  12  ft.  and  must  safely  support  a 
brick  wall  12  ins.  thick  and  51  ft.  high,  weighing  120  Ibs. 
per  cubic  foot. 

Weight  of  wall  =  12  X5J  X  120  =  7920  Ibs.  =3.96  tons. 

Then   -=  l.OOOTFL  =  1x3.96x12  =  47.52. 
c 

I  =0.084TFL2  -0.084  X  3.96  Xl22  =47.91. 

By  Table  21  a  box  lintel  8xl2xf  ins.  metal  will  be 
ample. 

PROBLEM  12.  Box  lintel  with  three  webs  supporting 
brick  wall.  Span  16  ft.  and  wall  24  ins.  thick  and  solid. 

If  the  lintel  be  shored  up  until  the  mortar  sets  properly, 
it  is  generally  assumed  that  the  volume  of  the  brick  wall 


24  SIMPLIFIED  FORMULAS  AND  TABLES 

actually  supported  by  the  lintel  is  that  included  below 
lines  drawn  at  60°  through  each  end  of  the  clear  span  of 

i  A 
the  lintel.     In  this  case  the   altitude  of  this  triangle  =  -~- 

tan  60°  =  13.86  ft. 

13.86X16.0X2 
Volume  of  brickwork  = ~ =  221. 76  cu.  ft. 

Weight  =  221.76X120  =  26611  Ibs.  =  13.30  tons. 
The  ordinary  formulas  given  for  such  a  mode  of  loading 


are 


PI     SI  PI* 

-z-  =  -       ana    A  = 


6     '  c  &OET 

Transforming  these  into  simplified  formulas  in  the 
manner  explained  in  Art.  4,  we  obtain  the  following  for- 
mulas for  this  form  of  loading  on  cast-iron  lintel: 

-  =  1.333  W  L    and    /  =  0.108  W  L2. 
c 

Then  -  =  1.333  WL  =  1.333x13.306X16  =  283.8. 
c 

I  =  0.108  WL2=  0.108 X13.306X162=  359.5. 

By  Table  23  a  lintel  12x24x1^  ins.  metal  will  suffice. 

PROBLEM  13.  Sheathing  of  roof.  Shortleaf  pine  J-in. 
thick.  Inclination  of  roof  35°.  Slated  on  felt  and  sheath- 
ing. 

p  =  10  Ibs.  (slates)  +  1  Ib.  (felt) +3  Ibs.  (sheathing)  = 
14  Ibs.  per  inclined  square  foot. 

s  =  15  Ibs.  per  horizontal  square  foot. 
s  cos  i°  =  12.3  Ibs.  per  inclined  square  foot. 

w  =  31.1   Ibs.    per    inclined    square     foot    (medium 
exposure). 


CAST-IRON  LINTELS  25 

Then  (14  +  12.3)  cos  35°  =  21.6  Ibs.  =  normal  component 
p+s  per  inclined  square  foot.  And 

14  cos  35°  +31.1  =  42.8  Ibs.  =  normal  component  p+w  per 
square  foot. 

14  sin  35°  +0.00  =    8.0  Ibs.  =  parallel  component  p+w  per 
square  foot. 

The  maximum  normal  component  =42.8  Ibs.  is  to  be 
taken,  and  the  parallel  component  8.0  Ibs.  may  be  neglected, 
because  resisted  by  edgewise  stiffness  of  the  sheathing. 
By  formulas  for  Case  5  b: 

38.3*      38.3X0.875 

L  =  —  -7=^  =  ~     /  --  —  -  =  5.06  ft.  on  centres  on  rafters. 
Vtc  V42.8 

12.1  1      12.1  xO.875 
L  =  —57=^  =  --  »/  --  —  =  3.02  ft.  on  centres  of  rafters. 

v  w  V42.8  . 

Hence  the  rafters  cannnot  be  placed  over  3  ft.  on  centres. 

PROBLEM  14.  Rafters.  Case  5  a.  Shortleaf  pine.  The 
rafters  of  the  same  roof  are  12.5  ft.  long,  and  their  weight 
averages  3  Ibs.  per  square  foot  of  inclined  surfaces. 

Then  p  =  14+3  =  17  Ibs.  per  inclined  square  foot  of  roof. 
(17  +  12.3)  cos  35°  =  24.0  Ibs.  =  normal  component  of  p  +s. 
17  cos  35°  +31.1     =45.0  Ibs.  =  normal  component  of  p  +w. 
17  sin  35°  +0.00     =   9.8  Ibs.  =  parallel  component  of  p+w. 
1.  Assume  that  rafters  are  set  3  ft.  on  centres. 


45X12.52X36 


8800  8800 

45X12.53X36 


21337  =          21337 


By  Table  19:  If  X  12,  2x10,  3x8,  4x8. 
By  Table  20:  If  X  12,  2x10,  3x10,  4x8. 


26  SIMPLIFIED  FORMULAS  AND  TABLES 

It  would  be  most  economical  to  use  2x10  rafters  if  full 
size.  But  these  would  look  heavy,  and  would  have  a  better 
appearance  if  set  closer  and  made  smaller. 

2.  Assume  a  section  If  X8  and  determine  c. 

By  Table  19:  -  =  17;  and  by  Table  20:   I  =  79,  for  this 
c 

section. 

By  formulas  for  e,  Case  5  a,  Table  8. 

7    8800        17x8800 
6  ==  ~c  *  wL2  =  45  X  12  52  =          ms*  on  cen^res 

213377      21337x79 
e  =    w  L3     =  "45x12  53  =          ms*  on  centres 

Best  use  If  -in.  rafters  set  18  ins.  on  centres. 


45X12.54X18 

rhen  A  =  6402007  =     640200X79     =  °'391  m'  =  max" 
imum  deflection. 

9.8XL5X12.5 
Also  --  o  v  2000  —  =  0*04"  ton  =  longitudinal  compres- 

sion at  mid-length  of  the  rafter  due  to  parallel  com- 
ponent of  its  load. 

And  pfTTo  =  0.028  ton  per  sq.  in.  compression  there. 

Then  0.028  (  1  +  ^r)  =  °-028  (  l  +6X°Q'391)  =  0.0239  = 

\        a  I  \  »      / 

maximum  fibre  stress  due  to  longitudinal  compression. 

Also  0.55  -  0.0239  =  0.526  =  maximum  safe  fibre  stress  for 
supporting  transverse  load  on  rafter. 

Substituting  this  value  in  the  general  formula  of  Case  5  a 
for  safety  against  rupture: 

7        wL2e     _  45X12.52X18  _ 
c  ~  16000  F  ":  16000X0.526  ~~  = 


CAST-IRON  LINTELS  27 

Since  this  is  less  than  the  actual  value  of  -=17  for 

If  X8  section,  this  size  will  amply  resist  both  normal  and 
parallel  components  of  loading. 

This  example  shows  that  in  ordinary  cases  the  parallel 
component  of  the  loading  on  rafters  may  be  neglected. 

PROBLEM  15.  Purlins  of  roof,  one  to  a  panel.  Length 
16  ft.,  set  normal  to  inclined  surface. 

1.  Assume  shortleaf  pine  timber,  average  weight  3  Ibs. 
per  inclined  square  foot. 

p  =  17+3  =20  Ibs.  per  inclined  square  foot. 

(20  +  12.3)  cos  35°  =  26.4  Ibs.  =  normal  component  of  p+s. 
20  cos  35°  +31.1    =  47.5  Ibs.  =  normal  component  of  p  +w. 
20  sin  35°  +0         =11.5  Ibs.  =  parallel  component  of  p  +w. 
16x12.5  =  200  sq.ft.  of  inclined  surface  supported 

by  one  purlin. 

W  =  200  X47.5  =  9500  Ibs.  =  4.75  tons  =  normal  loading 
on  purlin. 

W"=  200x11.5  =  2300  Ibs.  =  1.1 5  tons  =  parallel  loading 
on  purlin. 

a.  For  normal  loading  W. 

-  =  2.730  W  L  =  2.730  X4.75  Xl6  =  207.5. 
c 

7  =  1.125  WL2=  1.125  X4.75X162=  1368. 

By  Table  19:  4x18,  6x16,  8x14,  10x12. 
By  Table  20:  4x16,  6x14,  8x14,  10x12. 

b.  For  parallel  loading  W". 

^=2.730  WL=  2.730X1.15X16=  50.2. 
7  =  1.125TFL2  =  1. 125X1. 15X162=  331.2. 


28  SIMPLIFIED  FORMULAS  AND  TABLES 

By  Table  19:  18x6,  16x6,  14x6,  12x6,  10x6. 

By  Table  20:  18x8,  16x8,  14x8,  12x8,  10x8. 

Hence  8x14  might  suffice  for  the  dimensions  required 
by  both  loadings. 

Since  the  neutral  axis  of  the  cross-section  of  the  purlin 
cannot  coincide  with  its  minor  axis  in  this  case,  it  becomes 
necessary  to  determine  the  actual  maximum  fibre  stresses 
occurring  in  the  corners  most  distant  from  the  neutral  axis, 
by  the  formula  of  Art.  25. 

By  Table  20,  for  8x14  section,  Iv  =  1829;  for  14x8 
section,  Ix  =  597. 

4.75X141.15X8\ 

=0'622 


ton  per  square  inch  equals  actual  maximum  fibre  stress, 
which  exceeds  the  maximum  safe  fibre  stress  of  0.55 
ton  per  square  inch  for  shortleaf  pine. 

Hence  it  will  be  necessary  to  enlarge  the  section  of  the 
purlin,  say,  to  10x14. 

By  Table  20,  for  10x14,  Iv  =  2287;  for  14x10,  Ix  =  1167. 


Then  0.65Xl6>g7     +  '117       =0.449  ton  per  sq. 
in.,  which  is  amply  safe. 

2.  Assume  purlin  composed  of  two  latticed  steel  channels 
spaced  apart  to  make  purlin  equally  stiff  in  both  directions. 
Average  weight  of  steel  purlins  4  Ibs.  per  inclined  square  foot 
of  roof. 

Then          p  =  17  +4  =21  Ibs.  per  inclined  square  foot. 

21  cos  35°  +31.1  =  48.3  Ibs.  =  normal  component  of  p  +w. 

21  sin  35°  +0.00  =  12.0  Ibs.  =  parallel  component  of  p  +w. 
W  =  200  X48.3  =  9660  Ibs.  =  4.83  tons  =  normal  loading. 
W"  =  200  X  12.0  =  2400  Ibs.  =  1.20  tons  =  parallel  loading. 


CAST-IRON  LINTELS  29 

a.  For  normal  loading. 

-  =  0.187  WL  =0.187X4.83X16  =  14.4 
c 

I  =  0.047  W  L2  =  0.047  =  4.83  X 162  =  58.1. 

By  Cambria,  2,  8  in.,  11J  Ib.  channels  will  suffice. 

It  is  evidently  unnecessary  here  to  compute  -  and  /  for 

c 

the  parallel  loading,  since  their  values  are  much  smaller 
and  the  purlin  is  made  to  be  equally  stiff  in  both  directions. 

But  it  will  be  well  to  apply  the  formula  to  determine 
the  actual  maximum  fibre  stresses  occurring  in  the  section. 

Here  Iy  =  L  =  64.6,  and  b  =  9.43  ins.,  =  width  of  two 
flanges  +  spacing. 


tons  per  square  inch,  which  exceeds  the  maximum  safe 
fibre  stress  of  8  tons  for  steel. 

Hence,  the  purlin  must  be  composed  of  2,  8  in.,  13f  Ib. 
channels,  which  will  be  amply  strong. 

3.  Assume  that  purlin  is  composed  of  a  single  I-beam 
with  supporting  rods  as  required. 

Since  for  W,  -  =  14.4,  and  I  =  58.1,  as  already  found,  use 

V 

7      4  04 

1,  8  in.,  20 J  Ib.  I-beam,  for  which  sidewise  -  =  —^  =  1.98 

and  I  =  4.04.    Then  by  formulas  for  Case  5 : 

/    5.333  _  1.98X5.333  _  8.80  ft.  between   supporting 
L~  cX  W"   ''          1.20     "~      rods. 


/   T  /4  04 

L  =  4.64^^7?  =  4.64^:^0  =  10.22  ft.  between  rods. 


30  SIMPLIFIED  FORMULAS  AND  TABLES 

Therefore,  one  supporting  rod  at  mid-length  of  purlin 
will  suffice,  and  this  will  be  much  lighter  and  more  economi- 
cal than  the  latticed  purlin  composed  of  two  channels. 

Since  this  beam  is  supported  sidewise  at  the  middle  of 
its  length,  the  maximum  fibre  stress  at  that  point  is  only 
that  produced  by  W. 

Transposing  for  F  the  general  formula  in  Case  5: 

/_  1.5  WL 

c~       F     > 

we  find 

,,     1.5  WL      1.5X4.83X16 

r  =  — j —  = ^—      -  =  7.73  tons  per  square  inch, 


which  is  entirely  safe  there. 

Apply  formula  for  actual  maximum  fibre  stress,  the  free 
span  being  here  reduced  to  8  ft.  instead  of  16  ft.,  and  W' 
and  W"  are  likewise  halved. 

42x8  .    0.60X4.08 


60.2  4.04 

5.57  tons  per  square  inch,  which  is  amply  safe,  so  that 
this  I-beam  may  be  used. 


TABLES 


32 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE  1.  BEAM  CANTILEVER.  LOAD  AT  FREE  END.  TABLE  1 

General  Steel  Cast  Iron         Fir,  Wash.        Hemlock 

For  maximum  safe  fibre  stress  F. 


I     12WL 

c~     F 

1.5WL 

8.0TFL 

17.2TFL 

26.7TFL 

F  =  -> 

<m 

I  v  0.667 

cx~~T~ 

7     0.125 
7X~L~ 

7     0.058 

'  —  X       r 

c        L 

7      0.038 

—  X  —  f 
c         L 

T.-L. 

,J_ 

I     0.667 

I     0.125 

I     0.058 

I     0.038 

=  17280 


^ 


172801F 


r     7      1440LF 

•*  "~     ^       i? 
c          ^ 


For  maximum  safe  deflection  —  . 


1.192TFL2         2.160T7L2         24.70TFL2         38.45TFL2 


EI 

(\  Sd.O 

0463  7             I 

/ 

n  0°^ 

17280L2 

U.OlU  j  2 

0.463  L2 

0.041  L2 

n  909^  / 

J      El 

n  01  1-\  / 

n  fisru 

n  1«1  -*/ 

For  directly  computing  7  from  —  . 


-X0.795L       -X0.270L        -X1.44L 


-X1.44L 


For  maximum  safe  fibre  stress  and  deflection. 


1440^ 


576TFL' 


1.26c 


3.71c 


0.70c 


Actual  maximum  deflection. 
WL3  WL3  WL3 


25.207 


13.897 


1.2157 


0.70c 


WL*_ 
0.7827 


TABLES 


33 


CASE  1.     BEAM  CANTILEVER.     LOAD  AT  FREE  END.      TABLE  1 


Oak,  Wh.  Pine,  L.L.         Pine,  S.L.         Pine,  Wh. 

For  maximum  safe  fibre  stress  F. 


For  maximum  safe  deflection  — . 


=  23.1TFL2  20.4TFL2  28.8TFL2  34.6TFL2 


TF  =  0.043  7-,  0.049V.  0.035  •£ 


L-0.207^1  0.221^        0.187-y/^        °-17°V^ 


0.80c 


J^! 

:  1.327 


For  directly  computing  7  from  — . 


-X1.32L          -X1.30L 


For  maximum  safe  fibre  stress  and  deflection. 
0.84c  0.76c  0.77c 

Actual  maximum  deflection. 


WL* 
1.487 


WIS 
0.877 


Spruce 


-  =  18.5TFL 

17.2WL 

21.8TFL 

26.7T7L 

21.SWL 

~~c         L 

I_  X0.058 

I_    0.046 

7     0.042 
cX    L 

I     0.046 
c,X    L 

7     0.054 

7     0.058 

7     0.046 

7     0.042 

I  W0.046 

L     c        W 

cX    TF 

cX    W 

c  X    W 

cX    If 

26.6TFL2 


0.38 


-X1.22L 
c 


0.82c 


WL3 


1.137 


34 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE  2.     BEAM  CANTILEVER.     LOAD  UNIFORM.     TABLE  2 
General  Steel  Cast  Iron          Fir,  Wash.         Hemlock 

For  maximum  safe  fibre  stress  F. 


I    QWL 

G.75WL 

4. 

OOT7L 

8.58TFL 

13.33TFL 

c       F 

w-1  x- 

~CX6L 

I 

1.333 

7 

0.250 

7 

0.117 

7 

0.075 
L 

L 

c 

X    L 

c 

X    L 

c 

L        X 

I 
c 

1.333 

I 

0.250 

7 

0.117 

7 
c 

N/0.075 

c 

c 

A    W 

/\       JJ-T 

6480TFL2 


7      1080LF 

•*  /N  T^ 

C  Hi 


For  maximum  safe  deflection  — . 


6480L2 
A'648W 

U.t'K  W  Li 

2.240  ^2 
1.50^^ 

U.OJ-UKC  J-/-                 U.^UrK-L/ 

1.235  ^2            0.108  ^2 
1.11^           0.33^ 

J-^.^VJ  KK  i> 

0.069  L 
0.26^^ 

For  directly  computing  7  from  — . 

C 


-X0.596L        -X0.203L        -X1.08L          -X1.08L 

s*  /»/*/» 


c  c 

For  maximum  safe  fibre  stress  and  deflection. 


1.68c 


4.94c 


Actual  maximum  deflection. 


216TFL3 


WL* 
67.207 


JFL^ 
37.057 


JF7,_3 
3.247 


2.087 


TABLES 


35 


CASE  2.     BEAM   CANTILEVER.     LOAD  UNIFORM.     TABLE   2 
Oak,  Wh.  Pine,  L.L.         Pine,  S.L.         Pine,  Wh.  Spruce 

For  maximum  safe  fibre  stress  F. 


-  =  9.28TFL 


7     0.108 


=  8.65TFL2 


:0.116-^; 


7=-X0.938L 


8.54TFL 


10.91TFL 


13.33TFL 


10.91TFL 


7     0.117 

I     0.092 

7     0.075 

7     0.092 

cX    L 

cX    L 

cX    L 

cX    L 

I     0.117 

7     0.092 

7     0.075 

I     0.092 

cX    W 

cX    W 

cX    W 

cX    TT 

For  maximum 

safe  deflection 

L 
30' 

7.63  TFL2 

10.80TFL2 

12.96TFL2 

9.98TFL2 

0.131  L 

0.093  ^2 

0.077  ^2 

o.ioo  ± 

0-^W 

°-3WJ 

0.28VJ 

0.32^1 

For  directly  computing  7  from  — . 


-X0.891L       -X0.991L       -X0.973L     -X0.915L 
c  c  c  c 


=  1.07c 


WL* 


For  maximum  safe  fibre  stress  and  deflection 
1.12c  l.Olc  1.03c 

Actual  maximum  deflection. 


3.477 


JFL3 
3.947 


_ 
2.787 


WL? 
2.327 


1.09c 


WL3 
3.017 


36 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE  2A.     JOIST  CANTILEVER.     LOAD  UNIFORM.    TABLES 

General  Steel  Cast  Iron         Fir,  Wash.        Hemlock 

For  maximum  safe  fibre  stress  F. 


7_  wL*e 
c~4WOF 

32000 

6000 

wL2e 
2800 

1800 

I     4000*1 

7      32000 

7      6000 

7     2800 

7  X1800 

c       Lze 

c       7/2e 

c       L2e 

c  X  L2e 

c       L*e 

I     4000F 

"  c  X  wLz 

7      32000 
c       wT/2 

7      6000 
c       wL2 

7     2800 

I  X1800 

c       wL2 

c       wLz 

L  =  \\  —  X  
*  c         we 

178.9^  — 

77W^ 

52>9\^ 

42.4-y/— 

For  maximum  safe  deflection 

L 
30' 

wL3e 

wL% 

it?L3e 

wL3e 

wL'e 

3.70E 

53650 

29580 

2590 

1665 

3.70#7 

536507 

295807 

25907 

16657 

L3e 

L3e 

3.70#7 

536507 

295807 

25907 

16657 

wL3 

wL3 

wL* 

.  ^/     we 

37.7^/1 

*  UD(s 

30.9^e 

13.7^/1 

"Ws 

For  directly 

computing  7  from  —  . 

T    7  .  1080LF 

7  X0.596L 
maximum  safe 

-X0.203L 
fibre  stress  and  ( 

-X1.08L 
c 

leflection. 

-X1.UH, 

7   c  '     # 

For 

'        7?c 

1.68c 

4.94c 

0.93c 

0.93c 

1080F 

Actual  maximum  deflection. 

A      wL4e 

wL*e 

wL*e 

wL<e 

wL4e 

1609500 


888000 


77700 


49950 


TABLES 


37 


CASE  2A.    JOIST  CANTILEVER.     LOAD  UNIFORM.    TABLE  3 
Oak,  Wh.  Pine,  L.L.         Pine,  S.L.         Pine,  Wh.  Spruce 

For  maximum  safe  fibre  stress  F. 


I    wL2e 
c~2666 


7     2600 

—  X-yr- 
c      L2e 


7  v  2600 
e=—  X  —  YT 

c      wL2 


51 


2800 

7^     2800 
c       L2e 

I  .2800 


.oJ—  52.9A/— 

*  wee  *  wei 


wLze 
2200 


7     2200 

f\       T   n 

c       L2e 


7     2200 
cXwL2 


46.9 


.9A/— 
\  iyec 


1800 

7^     1800 
c       L2e 


7      1800 
c      wL* 


.  wL2e 
2200 

7  v  2200 


I     2200 
c 


42.4  J—         46.9  A/— 
'  wee  i  wee 


For  maximum  safe  deflection      . 


wL3e 

"2775 


w  = 


27757 
L3e 

27757 


3145 

wL3e 
2220 

wL3e 
1850 

wL3e 
2405 

31457 

22207 

18507 

24057 

31457 

22207 

~wL3 

L3e 
18507 

24057 

wL3 

wL3 

wL3 

we 


13 


.4^/1 
>  w;e 


=  1.07c 


_ 
~832~50 


For  directly  computing  7  from  — . 


-X0.890L       -X0.990L       -X0.973L 
c  c  c 


For  maximum  safe  fibre  stress  and  deflection. 
1.12c  l.Olc  1.03c 

Actual  maximum  deflection. 


94350 


66600 


55500 


-X0.9HL 
c 


1.09c 


wL*e 
72150 


38 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE  2s    FLOORING  CANTILEVER.      LOAD  UNIFORM.    TABLE  4 

General  Steel  Cast  Iron      Fir,  Wash.          Hemlock 

For  maximum  safe  fibre  stress  F. 


,       \wtf_ 

V667F  21.6  ~17.3~ 

=  667F  466.9  300.2 

L2  L2  ~L2~ 

/667W  2UH  17.31 

L~\      w  Vw 

For  maximum  safe  deflection  — . 

_  3/3.24WL3  L\/w 

~\       E  6.00  "5705" 

Et3  t3  /s 

216  F3  128.9  £ 

6.00^ 
>^ 

For  maximum  safe  fibre  stress  and  deflection. 

° 
Actual  maximum  deflection. 

wL4  wL*  wL* 
6482l» 


TABLES 


39 


CASE  2B.     FLOORING  CANTILEVER.     LOAD  UNIFORM.    TABLE  4 
Oak,  Wh.  Pine,  L.L.         Pine,  S.L.         Pine,  Wh.  Spruce 

For  maximum  safe  fibre  stress  F. 


t 

LV~w 

LVw 

LVw 

LVw 

LVw 

20.8J 

21.6 

19.2 

17.3 

19.2 

?/• 

433.6 

466.9 

366.9 

300.2 

-     366.9 

L2 

L2 

L2 

L2 

L2 

T 

20.8* 

21.6* 

19.2* 

17.3* 

19.2* 

'7Ti4 


L=— - 


For  maximum  safe  deflection  — . 


6.40  5.70 

262.5  j^  185.2  ~ 

6.40*  5.70* 


5.37 


154.3    - 


5.37^ 


For  maximum  safe  fibre  stress  and  deflection. 
0.56*  0.51*  0.52* 

Actual  maximum  deflection. 


6945* 


wL* 

7871*3 


_ 
5556*3 


wL* 
4630*3 


5.86 


20L5 


5.86* 


0.55* 


6019*3 


40 


SIMPLIFIED  FOEMULAS  AND  TABLES 


CASE   3.     BEAM 

CANTILEVER.     LOAD   IRREGULAR. 

TABLE   5 

General 

[Steel              Cast  Iron         Fir,  Wash. 

Spruce 

For  maximum  fibre  stress  F. 

7     12M 

c        F 

1.50M              8.00M              17.16M 

26.70M 

»-H 

0.667-              0.125-            0.058- 
c                        c                       c 

0.038- 
c 

For  maximum  safe  deflection  —  .     Load  at  free  end. 

,     17280ML 

1.193ML          2.160ML          24.70ML 
o  8d.o                n  d.fi^                n  041 

38.45ML 
0.026  j- 

E 
I/-     EI 

17280L 

Li                     ^    Li-                         Li 

o  QAO                n  Aft4?                n  OAT 

0.026^ 

17280M 

Af                     M                     M 

For  directly  computing  7  from  —  . 

7     1440LF 

-X0.795L       -X0.270L       -X1.44L 
c                        c                        c 

7cXl.44L 

Actual  maximum  deflection. 

576ML2 

ML2                  ML2                 ML* 

ML2 

EI                      25.207                13.897               1.2157 
For  maximum  safe  deflection  5^.    Load  uniform. 

0.7827 

T     12960ML 

0.894ML          1.620ML          18.52ML 
1.120  1             0.618  ^            0.054  ^ 

28.80ML 
0.035  ^ 

E 

M          EI 
12960L 

11  on   •*                 OfilR                     00^4 

/ 

"12960M 

M                      M                       M 

M 

For  directly  computing  7  from  —  . 

7     1080LF 

7X0.596L         7X0.203L        -X1.08L 
c                        c                        c 

-X1.08L 

i  —    X       j-, 

C              I'j 

Actual  maximum  deflection. 

432ML2 

ML2                 ML2                 ML2 

ML2 

EI 

33.607               18.537               1.627 

1.047 

TABLES 


41 


CASE   3.     BEAM   CANTILEVER.     LOAD   IRREGULAR. 

TABLE   5 

Oak,  Wh. 

Pine,  L.L.         Pine,  S.L.         Pine,  Wh. 

Spruce 

For  maximum  safe  fibre  stress  F. 

-  =  18.48M 

17.16M             21.84M             26.70M 

21.84M 

M  =  0.054  j 

0.058  -             0.046  -             0.038  - 
c                        c                        c 

0.046  - 

For 

maximum  safe  deflection  —  .     Load  at  free  end. 

/  =  23.07  ML 

20.36ML          28.85ML          34.60ML 

26.62ML 

M=  0.043  Y 

Li 

0.049  -£             0.035^             0.029^ 
LI                       LI                       LI 

0.038  y 
LI 

L=  0.043-^ 

0.049^            0.035^            0.029^ 

0.038  ^ 
M. 

For  directly  computing  7  from  —  . 

C 

7=^-Xl.25L 

-X1.19L         -X1.32L         -X1.30L 
c                       c                       c 

-X1.22L 
c 

Actual  maximum  deflection. 

A-ML2 

ML2                  ML2                  ML2 

ML2 

-1.327 

1.487                 1.047                0.877 

1.137 

For 

maximum  safe  deflection  —  .     Load  uniform. 

7  =  17.30ML 

15.26ML          21.60ML          25.92ML 

19.96ML 

M=  0.058  Y 
LI 

0.066  Y             0.047^             0.039^ 
LI                      Ju                      LI 

0.050  y 
Ju 

L=  0.058-^ 

0.066  -^            0.047  ^            0.039  ^ 
M                     M                     M 

0.050  £ 

For  directly  computing  7  from  —  . 

7=^X0.936L 

-X0.891L       -X0.991L       -X0.973L 

-X0.915L 

Actual  maximum  deflection. 

« 

A-ML2 

ML2                  ML2                  ML* 

ML2 

A~  1.747 

1.927                 1.397                 1.167 

1.517 

42 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE  4.  BEAM  SUPPORTED  AT  ENDS.  LOAD  AT  MIDDLE 

TABLE  6 


General 


7  = 


1080TFL2 


Steel  Cast  Iron         Fir,  Wash.        Hemlock 

For  maximum  safe  fibre  stress  F. 


I_3WL 

I75TFL 

,2.670 

<~T 

2.670 

2.000PFL 

7X~ZT 

7     0.500 

4.29TFL 
7     0.233 

6.67TFL 
7  0.150 

c       F 

v=T-x-              -: 

c     37/                     c 

L=LXJL            L> 

c        L 
I     0.233 

c  L 
7  0.150 

(    W 

c  X    W 

c  X    W 

c  X  W 

For  maximum  safe  deflection  — . 


0.135JFL2         1.544WL2         2.403T7L2 


1080L 


1080TF 


7     360LF 


13.430  f-2          7.410  f-2 
i>  A/ 


0-648 


For  directly  computing  7  from  — . 


0.417 


i    °-W^    o-W^ 


-X0.199L 

c  c 


-X0.360L      -X0.360L 
c  c 


EC 
~~3GOF 


36TTL3 
El 


For  maximum  safe  fibre  stress  and  deflection. 

5.04c  14.82c  2.78c  2.78c 


Actual  maximum  deflection. 

WL3  WL*  WL* 

4037  2227  19.457 


TFL3 
12.507 


TABLES 


43 


CASE    4.     BEAM    SUPPORTED    AT    ENDS.     LOAD    AT    MIDDLE. 

TABLE  6 


Oak,  Wh.  Pine,  L.L.         Pine,  S.L.         Pine,  Wh. 

For  maximum  safe  fibre  stress  F. 


Spruce 


±  =  4.61TFL 
c 


w=Lx^l 


L-7X°-217 

L-^x~w~ 


4.29TFL 

7  v,0.233 

—  X     f 
c         L 

7     0.233 
cX    W 


5A5WL 


7^0183 

7X~I7~ 


I     0.183 
cX    W 


6.67TFL 

7     0.150 
7X~L~ 


7      0.150 
c  X    W 


5.45TFL 


I     0.183 
7X    L 


-X 
c 


0.183 


For  maximum  safe  deflection     r. 


1.442TFL2 


=  0.695^ 


1.272TFL2         1.802T^L2         2.163TFL2 
0.787  ^0  0.555  ~  0.463  ^0 


1.965TFL2 
0.602  ^0 


.887^         0.745^         0.681^       0.776^ 


7=-X0.312L 


For  directly  computing  7  from  — . 

C 


-X0.296L       -X0.330L       -X0.324L      -X0.330L 
c  c  c  c 


3.20c 


For  maximum  safe  fibre  stress  and  deflection. 
3.37c  3.03c  3.09c 

Actual  maximum  deflection. 
WL3  WL*  WL* 


20.847 


23.637 


WL* 
16.677 


13.887 


3.29c 


WL* 
18.057 


44 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE   5.     BEAM   SUPPORTED   AT   ENDS.     LOAD   UNIFORM 

TABLE  7 


General 


7  =  1.5TFL 
c~     F 


W-i-X-?- 
~c  A1.5L 


T    •*      \* | 

cX1.5TF 


7  = 


675TFI/2 
E 

El 


1   El 
\675FF 


Steel  Cast  Iron          Fir,  Wash.        Hemlock 

For  maximum  safe  fibre  stress  F. 

0.187T7L          l.OOOTFL          2.144TFL          3.336TFL 

7  _  0.300 


21.50^  11.86  ri  1.22  ri 


4.64 


7     0.300 
cX    TF 


7     5.333          ^VLOOO          ^V0^67 
cXL  cXL  cXL 


7^    5.333          7     1.000          7     0.467 
cX    W  cX    W  cX    W 


For  maximum  safe  deflection  — . 

oU 


0.047TFL2         0.084TFL2         0.965TFL2         1.500TFL2 


0.67 


EC 


For  directly  computing  7  from  — . 


-X0.248L       -X0.085L       -X0.450L       - 
c  c  c  c 


For  maximum  safe  fibre  stress  and  deflection. 

4.02c  11.90c  2.22c  2.22c 


Actual  maximum  deflection. 


A  = 


22.5TFL» 
El 


WL* 
5807 


WL* 
3567 


WV 
31.17 


WL* 
20.07 


TABLES 


45 


CASE   5.    BEAM   SUPPORTED   AT   ENDS.      LOAD   UNIFORM 

TABLE  7 


Oak,  Wh.  Pine,  L.L.         Pine,  S.L.         Pine,  Wh. 

For  maximum  safe  fibre  stress  F. 


Spruce 


-  =  2.310TFL 

c 


L_7     0.433 
X 


2.144TFL 


2.730TFL          3.336TFL 


W 


2.730TFL 


7     0.467          ^V0^67         _/    0.300          7     0.367 

^N/-  y.^7  ^  T  ^7 

c         L  c         L  c        L  c         L 


7     0.467          7     0.367          7    0.300          LV0<367 

^\  TTT  _       /\  TTT  ^N  TTT  /X " 


W 


For  maximum  safe  deflection  ^:. 


=  0.900TFL2  0.795TFL2         1.125TFL2         1.350PTL2         1.038TFL2 


1.26 


0.89 


0.73 


0.96 


0.98A  ~ 


7=-X0.390L 
c 


For  directly  computing  7  from  — . 


-X0.370L       -X0.412L       -X0.405L       -X0.381L 
c  c  c  c 


L=2.56c 


For  maximum  safe  fibre  stress  and  deflection. 

2.70c  2.42c  2.47c 

> 

Actual  maximum  deflection. 


33.37 


WL* 
37.87 


26.77 


WIS 
22.27 


2.63c 


WL* 
28.97 


46 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE   SA.    JOIST   SUPPORTED   AT   ENDS.    LOAD   UNIFORM 

TABLE   8 


General 


Steel 


Cast  Iron         Fir,  Wash.          Spruce 


For  maximum  safe  fibre  stress  F. 


1       wL2e 

wL2e 

wL2e 

wL*e 

wL*e 

c     mOOF 
7      16000F 

128000 
7      128000 

24000 
7      24000 

11200 
7  X11200 

7200 
7  x  7200 

7      16000F 

c        L2e 
I      128000 

c       L2e 
7     24000 

7      11200 

cX  L2e 
7     7200 

"c        «L« 

c        wL2 

c       »L» 

c  ^  wL2 

c      wL* 

//      16000F 

357V^ 

For  maximu 

wL3e 

15fr%/  7 

10fU/   7 

84.9  V  — 
\w;ec 

A/  —  A/       A 

'  c       .  we 
j      wL3e 

lODA/ 

*  1VCC 

.m  safe  deflection 

1UOAI 

L 

30' 

35.56^7 

515620 
5156207 

284480 
2844807 

24927 
249277 

16000 
160007 

L*e 
35.56^7 

5156207 

wL3 

L3e 

2844807 

L3e 
249277 

L3e 
160007 

wL' 

wL3 

wL3 

wL3 

r      s/35.56^7 

80.2-v  — 
For  directly 

Vfl  94.87^ 

65.8^          29.2^ 

*  we                   *  iye 

computing  7  from  —  . 

3/7 

*      we 
7     450LF 

>w;e 
VO  d^OT 

cX     # 

For 

C                                C                                C 

maximum  safe  fibre  stress  and  deflection. 
4.02c                 ll.QOc               2.22c 

Actual  maximum  deflection. 

C 

2.22c 

L    450F 
ty7/4e 

1067EI 


154715007        85360007 


7469007 


4801507 


TABLES 


47 


CASE   SA.     JOIST   SUPPORTED  AT  ENDS.     LOAD  UNIFORM 

TABLE   8 


Oak,  Wh.  Pine,  L.L.         Pine,  S.L.         Pine,  Wh. 

For  maximum  safe  fibre  stress  F. 


Spruce 


7     wL2e                       wL2e~ 

wL2e                 wL2e 
8800                  7200 

wL2e 

^"10400                      11200 

8800 

7      10400               7      11200 

7      8800           7      7200 

7      8800 

7      10400               7      11200 

S\*       T  n                                        '  S\       T  n 

c       Li*e             c       L**e 
I     8800           7      7200 

c  X  L2e 
I     8800' 

e'cXwL2                 cX  wL2 

c      wL2            c      wL2 

93.8A/—         84.9-\/  — 
*  wee                 ~  wee 

93.8A/  — 
>  wee 

For  maximum  safe  deflection  —  . 

wL3e                      wL3e 

wL3e                  wL3e 

w;L3e- 

26670                      29606 

21337                17780 

23114 

266707                    296067 

213377              177807 

231147 

266707                    296067 

L3e                    L3e 

213377              177807 
wLi3                  wL3 

27.7jl±           26.1-4/1 

'  we                  V  we 

L3e 
231147 

wL3                         wL3 
L  =  29.9-\f—                30.9^/— 

wL3 
28.5^.' 

For  directly 

computing  7  from  —  . 

C 

7=-X0.390L             -X0.370L 
c                              c 

-X0.412L        -X0.405L 

-X0.381L 

For  maximum 

safe  fibre  stress  and  deflection. 

L=2.56c                      2.70c 

2.42c                 2.47c 

2.63c 

Actual  maximum  deflection. 

wL4e                       wL4e 

wL*e                  wL*e 

wL*e 

8002507 


9069507 


6402007 


5335007 


6402007 


48 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE  SB.    FLOORING  SUPPORTED  AT  ENDS.     LOAD  UNIFORM 

TABLE  9 

General                       Steel            Cast  Iron         Fir,  Wash.  Hemlock 
For  maximum  safe  fibre  stress  F. 

I  wLz                                                                      L/Vw  LVw 

=  \2667F                                                                   ~43ir  ~34lT 

2667F*2                                                                   1867*2  1200*2 

w  =  -&-                   -JT  "IT- 

r  _    / 2667/^*2                                                                 43.2*  34.6* 

Ll~-\—  7=r  ==• 

W                                                                                                      -\/w  -\/w 

For  maximum  safe  deflection  ^r. 

t=i2.96E                                                                    14.4  TOT 

2  96-&*$                                                                      2072*3  1332*8 

W  =  -VT-                     ~V~  ~~IJ~ 

L=  8/2.96#<3                                                                 14.4*  11. Ot 

'      iy                                                                     -\/w  "Vw 

For  maximum  safe  fibre  stress  and  deflection. 

L=<jjp 1.06*  1.11* 

Actual  maximum  deflection. 

wL*                                                                           wL*  wL4 

T067J^*3                    746900*3  480150*3 


TABLES 


49 


CASE  SB.    FLOORING  SUPPORTED  AT  ENDS.    LOAD  UNIFORM 

TABLE  9 


Oak,  Wh. 


*  = 


L  = 


41.6 
1734*_2 

41.6* 

Vw 


Pine,  L.L.         Pine,  S.L.         Pine,  Wh. 
For  maximum  safe  fibre  stress  F. 


43.2 

JL867*2 
L2 

43.2* 

Vw 


38.3 

1467*_2 
L2 

38.3* 


34.6 

1200*2 
L2 

34.6* 


Spruce 


38.3 

^1467*2 
L2 

38.3* 


~  13.1 


2220*3 


_13.1* 

•k  —        3/~ 


=  1.28* 


For  maximum  safe  deflection  — . 

30 


13.6 

Lv^ 
12.1  ' 

11.4 

12.5 

"T3" 

1776*3 
L3 

1480*3 
L3 

1924*3 
L3 

13.6* 

12.1* 

11.4* 

12.5* 

For  maximum  safe  fibre  stress  and  deflection. 
1.35*  1.33*  1.24* 

Actual  maximum  deflection. 


wL* 


800250*3 


906950*3 


640200*3 


wL* 
533500*3 


1.32* 


wL* 
693550*3 


50 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE    6.     BEAM    SUPPORTED    AT    ENDS.      LOAD    IRREGULAR 

TABLE  10 


General 


I    12M 

c       F 


Steel  Cast  Iron         Fir,  Wash. 

For  maximum  safe  fibre  stress  F. 

1.50M  8.00M  17.15M 


-X  0.667 
c 


^-XO.125 


-X  0.058 


Hemlock 


26.70M 


-X  0.038 
c 


For  maximum  safe  deflection  — .     Load  at  middle. 


4320ML 


E 

M  =  4320L 

3.360  -^ 

1.850  y 
L/ 

0.162  y 

L/ 

0.104  y 
L/ 

EI 

Q  Q  AH    ^ 

/ 

1  S'lO 

0.162^ 

0.104^ 

4320M 

O.OvMJ    »  *• 

M 

M 

A=- 


7  = 


E 

144ML2 
EI 

5400ML 


For  directly  computing  7  from  — . 

-X0.199L         -X0.068L        -X0.360L 
c  c  c 


Actual  maximum  deflection. 


ML2 
100.77 


ML2 
55.67 


ML2 
4.867 


For  maximum  safe  deflection.     Load  uniform. 


-X0.360L 


ML2 
3.137 


E 

U.O/^lW  JL/ 

U.tUOlK£.L/ 

4  .t  A1V1LI 

\.£.  \J\J1V1  Li 

EI 
:5400L 

2.68  y 
LI 

1.48y 
Li 

0.130-^ 

0.083  T 
LI 

EI 

I 

I 

M 

/ 

f)  f|QO     "* 

5400M 

M 

M 

7     450LF 


A  = 


180ML2 
EI 


For  directly  computing  7  from  — . 

-X0.248L        -X0.085L        -X0.450L 

Actual  maximum  deflection. 
ML2  ML2  ML2 


80.77 


44.47 


3.897 


3.127 


TABLES 


51 


CASE    6.    BEAM    SUPPORTED    AT    ENDS.     LOAD    IRREGULAR 

TABLE   10 


Oak,  Wh.  Pine,  L.L.        Pine,  S.L.         Pine,  Wh. 

For  maximum  safe  fibre  stress  F. 


-  =  18.48M 
c 

17.15M 

21.84M 

26.70M 

M  =  0.054  - 

0.058  - 
c 

0.046  - 
c 

0.038  - 

For 

maximum  safe 

deflection  5^. 

Load  at  middle. 

7  =  5.77  ML 

5.08ML 

7.21ML 

8.64ML 

M  =  0.174  4 
Li 

0.197  4 
LI 

0.139  4 
LI 

0.116  4 
Li 

L  =  0.174^ 

0.197^ 

0.139^ 

0.116  ^ 

Spruce 


21.84M 


0.046  *- 


6.65ML 

0.151  4 
LI 


For  directly  computing  7  from  —  . 


7=-X0.312L 


-X0.297L       -X0.330L       -X0.324L      -X0.330L 
c  c  c  c 


5.217 

7.20ML 
7 


M  =  0.139 


Actual  maximum  deflection. 
ML2  ML2  ML2 


5.907 


4.177 


3.477 


For  maximum  safe  deflection.     Load  uniform. 


6.35ML 


9.00ML 


10.80ML 


ML2 
4.527 


8.32ML 


0.158  4 
LI 

0.158  L 

0.111  y 

Ll 

°-m-5TF  ! 
M 

0.093-^ 
0.093^ 

/ 
L 

For  directly  computing  7  from  —  . 

C 


4.177 


Actual  maximum  deflection. 
ML2  ML2  ML2 


4.727 


3.337 


2.787 


-  X0.382L 


ML2 
3.617 


52 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE  7.     BEAM  FIXED  AND  SUPPORTED 
MIDDLE.     TABLE   11 

AT  ENDS. 

LOAD  AT 

General 

Steel 

Cast  Iron 

Fir,  Wash. 

Hemlock 

For  maximum 

safe  fibre  stress  F  . 

/ 

2.25TFL 

0.281TFL 

1.50T7L 

3.20TFL 

4.98TFL 

c 

F 

W 

7x    F 

7     3.560 

7     0.667 

7     0.311 

7  .0.200 

'    c  A2.25L 

cX    L 

cX    L 

cX    L 

c        L 

L 

7  v     F 

I     3.560 

7     0.667 

7     0.311 

7     0.200 

c  X2.25L 

cX    T7 
For  maximum 

c        W 
safe  deflection 

cX    TF 

L 
30' 

cX    W 

I 

472.5T7L2 

0.0328TFL2 
3.070  — 

0.0591TFL2 
1.6941 

0.675TFL2 
1.482  ^2 

1.050TFL2 
0.953  1 

E 
=  472.5L2 

L 

I—El" 
\472.5TF 

'Wl 

1.30^ 

"WJ 

oW! 

210^ 


A=- 


15.75TFL* 
El 


For  directly  computing  7  from  —  . 


-X0.160L       -X0.394L       -X0.210L 
c  c  c 


For  maximum  safe  fibre  stress  and  deflection. 
8.66c  25.40c  4.77c 

Actual  maximum  deflection. 
TTL3  TFL3  WL3 


-X0.210L 


4.77c 


920.77 


508.07 


44.57 


28.67 


TABLES 


53 


CASE  7.     BEAM  FIXED  AND  SUPPORTED  AT  ENDS.     LOAD  AT 
MIDDLE.     TABLE  11 


Oak,  Wh.  Pine,  L.L.        Pine,  S.L.         Pine,  Wh. 

For  maximum  safe  fibre  stress  F. 


Spruce 


3.20WL 


4.08TFL 


4.98WL 


4.08T7L 


W 

I 

^0.289 

I 

0.311 

I     0.245 

I 

0.200 

I     0.245 

c 

X    L 

c 

X    L 

cX    L 

c 

X    L 

cX    L 

T 

I 

0.289 

I 

0.311 

7  > 

^0.245 

I 

0.200 

7  N 

0.245 

c 

X    W 

c 

X    W 

c  ' 

<    W 

c 

X    W 

c  ' 

K    Tf 

For  maximum  safe  deflection      . 


7=0.630WLa 


W  =  1.588 


0.556WL2         0.788  WL2         0.946  WL*         0.728  TFL2 
7 


1.800 


1.270 


1.058 


5.50c 


A  = 


WL* 
47.67 


For  directly  computing  /  from  — . 


-X0.181L       -X0.203L       -X0.198L 
c  c  c 


For  maximum  safe  fibre  stress  and  deflection. 
5.79c  5.20c  5.30c 

Actual  maximum  deflection. 

WL*  WL* 


TfL3 
53.07 


-X0.186I- 
c 


5.63c 


WL* 


37.17 


31.87 


41.37 


54 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE    8.     BEAM    FIXED    AND    SUPPORTED    AT    ENDS.     LOAD 
UNIFORM.    TABLE  12 

General  Steel  Cast  Iron         Fir,  Wash.       Hemlock 

For  maximum  safe  fibre  stress  F. 


7     1. 

5WL 

0.187TFL 
7     5.33 

l.OOOTFL 
7     1.00 

2.14TFL 
7  ^.0.467 

3.33IFL 
7  N  0.300 

c 
IF 

F 
X/0.667F 

c 

L-7 

X     L 
VX0.667F 

cX  L 

7     5.33 
cX  JF 

cX 
71 

L 

1.00 

cX    L 
7     0.467 

c         L 
7     0.300 

Tf 

c 

FT 

cX    TF 

cX    W 

270TFL2 


7      180LF 
cX     E 


For  maximum  safe  deflection  — . 

oO 


^ 

ir=  E/ 

U.UJ-OOKKl^- 

53.807 
L2 

7  9^-%  / 

u.uooo  kr  x>" 
29.657 

2.597 

1.677 

270L2 

L2 

54K/T 

L2 

L2 
L29\W 

L     J    EI 

L     \270PF 

/.dd'Y  yy 

5.44^^ 

L61A/TT 

For  directly  computing  7  from  — . 


-X0.099L       -X0.0347,       -X0.180L      -X0.180L 


1SOF 


For  maximum  safe  fibre  stress  and  deflection. 
lO.lc  29.7c  5.56c 


5.56c 


Actual  maximum  deflection. 


EI 


_ 
16127 


WL* 

8857 


JFL* 

77.87 


50.07 


TABLES 


55 


CASE  8.     BEAM  FIXED  AND  SUPPORTED  AT  ENDS.     LOAD 
UNIFORM.     TABLE  12 


Oak,  Wh.  Pine,  L.L.         Pine,  S.L.         Pine,  Wh. 

For  maximum  fibre  stress  F. 


Spruce 


*  =  2.31WL 

2.14TFL 

2.72WL 

3.33T7L 

2.72TFL 

w  J  x°-438 

7  ^0.467 

I  ^0.367 

7     0.300 

7  W0.367 

CX    L 

cX    L 

cX    L 

cX    L 

cX    L 

I     0.438 
~^X~~W 

I     0.467 
c  X    W 

7     0.367 
cX    TF 

^0.300 

7     0.367 
cX    W~ 

For  maximum  safe  deflection  —  . 
30 

7  =  0.360TFL2 

0.318TFL2 

0.450TFL2 

0.540WL2 

0.416TFL2 

w   2'7SI 

3.157 

2.227 

1.857 

2.417 

'    L* 

L2 

L2 

L2 

L2 

For  directly 

computing  7  from  —  . 

7=-X0.156L 

<j 

-X0.148L 

/ 

/ 

-X0.152L 

C 

For 

maximum  safe 

fibre  stress  and 

deflection. 

L  =  6.42c 

6.75c 

6.07c 

6.18c 

6.57c 

Actual  maximum  deflection. 

A  —         - 

WL* 

WL* 

TFL» 

TFL' 

83.37 

94.47 

66.77 

55.67 

72.27 

56 


SIMPLIFIED  FOEMULAS  AND  TABLES 


CASE  SA.    JOIST  WITH  ENDS  FIXED  AND  SUPPORTED.    LOAD 
UNIFORM  .   TABLE  13 

General  Steel  Cast  Iron         Fir,  Wash.       Hemlock 

For  maximum  safe  fibre  stress  F. 


I      wLze 

wL2e 

wL2e 

wL2e 

wL2e 

I    16000^ 

128000 
7     128000 

24000 
7    24000 

11200 
7^11200 

7200 
7     7200 

W~cX    L*e 

cX    L*e 
7     128000 

c  X  L*e 
I    24000 

c        Lre 
7X11200 

7    7200 
c       wL2 

84.9A/— 
^  wee 

TX    wL* 

358A/—          155V  — 
*  wee                  \  wee 

For  maximum  safe  deflection 
wL3e                 wL3e 

106  V  — 

L 
30' 

,VFT 

7~88.89# 
88.89^7 

1288905 
12889057 

711120 
7111207 

62223 
622237 

40000 
400007 

L3e 

88.89^7  . 

12889057 

7111207 

L3e 
622237 

L3e 
400007 

wL3 

wL3 

wL3 

wL3 

wL3 

L     3/88.89£7 
^     ISOLf; 

108.8A/  — 
'  we 

For  directly 
-X0.099L 

89.3A/—          39.6A/— 
'  we                   '  we 

computing  7  from  —  . 
-X0.034L       -X0.180L 

•urfZ 

>  we 
-X0.180L 

Ec 


For  maximum  safe  fibre  stress  and  deflection. 
lO.lc  29.7c  5.56c 

Actual  maximum  deflection. 


A  = 


wL4e 
2667^7 


5.56c 


3867150C7    213360007    18669007    12001507 


TABLES 


57 


CASE  SA.     JOIST  WITH  ENDS  FIXED  AND  SUPPORTED.     LOAD 
UNIFORM.     TABLE   13 


Oak,  Wh. 


I     wL*e 


Pine,  L.L.        Pine,  S.L.         Pine,  Wh. 
For  maximum  safe  fibre  stress  F. 


For  maximum  safe  deflection  57;. 

oU 


Spruce 


c     10400 

7      10400 
'~c  X  L2e 

11200 
7      11200 

8800 

7     8800 
c       Z/2e 

7200 
7      7200 

8800 

7     8800 
c  XL2e 

c        L2e 

I      10400 

7      11200 

7     8800 
c       wL2 

7     7200 

/_    8800 
c       w;7/2 

e  —     X      T  9 
c        t&L2 

c  X  wL* 

c      wL^ 

|~7~ 

L  =  102V  — 
"  wee 

106  V  — 
*  -u;ec 

93.8V  — 
'  i^ec 

84.9V  — 

93.8V  — 
^  wee 

tyL3e 

w;L3e 

44445 
444457 

wL3e 

66668 
666687 

75557 
755577 

53334 
533347 

57779 
577797 

L3e 
666687 

755577 

L3e 
533347 

L3e 
444457 

577797 

*-«Ws 

wL3 
42^/1 

*    1(^6 

wL* 

35.4^/1 

'  we 

wL3 

For  directly  computing  7  from  — . 

C 


7=— X0.156L       — X0.148L       — ; 


—  X0.162L       — X0.152L 
c  c 


6.42c 


A  = 


wL4e 
20002507 


For  maximum  safe  fibre  stress  and  deflection. 
6.75c  6.07c  6.18c 

Actual  maximum  deflection. 
wL4e  wL*e  wL4e 


22669507 


16002007 


13335007 


6.58c 


wL4e 
17335507 


58 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE   SB.      FLOORING    WITH    ENDS    FIXED    AND   SUPPORTED 
LOAD  UNIFORM.     TABLE  14 


General 


Steel  Cast  Iron         Fir,  Wash.        Hemlock 


For  maximum  safe  fibre  stress  F. 

I  wL*  kVw 

~\2667^  43.2 

_  26677^                                                                        186772  1200^ 

:    L2                                                                              L2  L2 

/2667F                                                                   43.2£  34.6Z 

/=\  --                         .........             .........            —  -7=  —  r=r 

\      w                                                                       Vw  ^w 
For  maximum  safe  deflection  ^. 

s/  wL3                                                                    L^/w  LL^™. 

17.3  15.0 

3335«3 

~7T 

173t  15.0J 

^w  Vw 

For  maximum  safe  fibre  stress  and  deflection. 

......  —'••-'     2-78(  2-78' 

Actual  maximum  deflection. 

wL*  wL* 
9990CH3 


TABLES 


59 


CASE   SB.     FLOORING   WITH    ENDS    FIXED   AND   SUPPORTED 
LOAD  UNIFORM.     TABLE  14 


Oak,  Wh. 


17.7 


17.7* 

**  ~        3/ 


L  =  3.21* 


Pine,  L.L.        Pine,  S.L.         Pine,  Wh. 
For  maximum  safe  fibre  stress  F. 


For  maximum  safe  deflection  ^. 


18.5 
6299^ 


16.4 
4446  ^ 


15.5 
3705  |^ 
15.5* 


For  maximum  safe  fibre  stress  and  deflection. 
3.38*  3.04*  3.09* 

Actual  maximum  deflection. 
wL*  wL*  wL* 


Spruce 


LVw 


41.6 

43.2 

38.3 

34.6 

38.3 

1734*2 

1867*2 

1467*2 

1200*2 

1467*2 

L2 

L2 

L2 

L2  j 

L2 

41.6* 

43.2* 

38.3* 

34.6* 

38.3* 

16.9 


4817  r* 


16.9* 


3.04* 


166500*3 


133200Z3 


111000*3 


144300*3 


60 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE  9.    BEAM  FIXED  AT  ENDS.    LOAD  AT  MIDDLE 
TABLE   15 


General. 


I  =  l.5WL 
c~     F. 


ri7_7  X/0.667F 

rr   —      /\         y 


L  = 


I  ..0.667F 


7  = 


W  = 


L  = 


270TFL2 


270JF 


7^    180LF 

•*    —         ^^  77T 


Steel.  Cast  Iron         Fir,  Wash.        Hemlock 

For  maximum  safe  fibre  stress  F. 


0.187TFL 
I     5.325 


I     5.325 
c  X    W 


l.OOTFL 
7     1.000 

/\          f 

c         L 


I     1.000 
c  X    W 


2.14TFL 
7     0.467 


I     0.467 
c  X    W 


3.33TFL 
7     0.300 


I     0.300 
c  X    W 


For  maximum  safe  deflection  ™. 


E 
El 

53.77 
L2 

29.67 
L2 

2.597 

270L2 

L2 

L677 
L2 


^Vw    «Ww 


For  directly  computing  7  from  — . 

c 


-X0.099L        -X0.034L        -X0.180L        -X0.180L 
c  c  c  c 


EC 
1SQF 


A  = 


9TFL3 
El 


For  maximum  safe  fibre  stress  and  deflection. 
lO.lc  29.7c  5.56c 

Actual  maximum  deflection. 
WL3  WL3 


16117 


888.77 


77.87 


5.56c 


WL3 
50.07 


TABLES 


61 


CASE  9.    BEAM   FIXED  AT  ENDS.    LOAD  AT  MIDDLE 
TABLE  15 


Oak,  Wh.  Pine,  L.L.        Pine,  S.L.         Pine,  Wh. 

For  maximum  fibre  stress  F. 


Spruce 


-  =  2.31TFL                  2.14TFL 

c 

II7    I  v/0.433                7  ^0.467 

2.72WL 
I     0.367 

3.33TFL 
7     0.300 

2.72TFL 
7     0.367 

~c  X    L                   c         L 
I     0.433                7  .,0.467 

c         L 
I     0.367 

c        L 
7  VX0.300 

7     0.367 

c  X    Tf                  c  X    W 

c        W  " 

c  X    TF 

0.360TFL2 


For  maximum  safe  deflection  ^. 


0.318T7L2 

3.157 
L2 


0.450TFL2 
2.227 


0.540TFL2         0.416FL* 


1.857 
L2 


2.417 
L2 


7=-X0.156L 


For  directly  computing  7  from  — . 


-X0.148L       -X0.165L       -X0.162L      -  X0.152L 

C  C  CO 


For  maximum  safe  fibre  stress  and  deflection. 
=  6.42c  6.75c  6.07c  6.18c  6.58c 

Actual  maximum  deflection. 

TFL3  WL3  WL*  WL3  WL3 

=  83^7  94^57  66^77  55.67  72^27 


62 


SIMPLIFIED  FOEMULAS  AND  TABLES 


CASE  10.     BEAM  FIXED  AT    ENDS.     LOAD  UNIFORM 
TABLE   16 


General 


W 


135TFL2 
~^~ 

EI 


135L2 

•^ 


135JF 


Steel  Cast  Iron          Fir,  Wash.        Hemlock 

For  maximum  safe  fibre  stress  F. 


7 
c 

WL 

F 

0. 

125TFL 

0.667TFL 

1. 

43T7L 

2.22WL 

V 

I      F 

7 

8.00 

7     1.50 

7 

0.70  • 

I     0.45 

\ 

~c  XL 

c 

X  L 

c        L 

c 

L 

c        L 

L 

=I^xw 

7 

c 

8.00 
A  JF 

7      1.50 
c  X  TF 

7^ 

c 

0.70 
X  W 

I     0.45 
c  X  W 

For  maximum  safe  deflection      . 


0.0093T7L2       0.0169TFL2       0.193TFL2         0.299TFL2 


107.4  -A 


59.3  ^n 


5.18  -f-n 


•35Vi         7.70^ 


3.33 


1.83^^ 


135F 


4.5TFL3 
EI 


For  directly  computing  I  from  — . 

C 


-X0.075L       -X0.025L       -X0.135L       -X0.135L 
c  c  c  c 


For  maximum  safe  fibre  stress  and  deflection. 
13.43c  39.50c  7.40c 


Actual  maximum  deflection. 
WL3  WLZ  WL3 


32257 


WL3 
17787 


155.57 


7.40c 


_ 
100.07 


TABLES 


63 


CASE  10.     BEAM  FIXED  AT  ENDS.     LOAD  UNIFORM 
TABLE  16 


Oak,  Wh.  Pine,  L.L.         Pine,  S.L.         Pine,  Wh. 

For  maximum  safe  fibre  stress  F. 


Spruce 


-  =  1.54TFL 

1. 

43TFL 

I.S2WL 

2.22TFL 

1.82TFL 

V=^X~T~ 

7 

c 

0.70 
X  L 

I     0.55 
c  X  L 

-X—  - 

7  0.55 
cX  L" 

L-fx9r 

7 

X—  - 

7     0.55 
c  X  TF 

7     0.45 
c  X  W 

7  0.55 
c  X  W 

For  maximum  safe  deflection  — . 

oU 


=  0.180TFL2 

0.159TFL2 

0.224  TFL2 

0.269TFL2 

0.208TF 

/ 

6.30  L 

4  44  — 
L2 

3.70  ^2 

4.82  1 

For  directly  computing  7  from  — . 


;7  =  ^-X0.117L  ^XO.lllL     .  ^X0.124L       ^X0.122L       -X0.114L 

For  maximum  safe  fibre  stress  and  deflection. 

L  =  8.55c  8.98c  8.10c  8.24c  8.76c 

Actual  maximum  deflection. 


186.77 


_ 
189.07 


133.37 


_ 
111.07 


WL3 
144.57 


64 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE  10A.     JOIST  WITH  ENDS  FIXED.     LOAD  UNIFORM 
TABLE   17 


General 


Steel 


Cast  Iron         Fir,  Wash.        Hemlock 


For  maximum  safe  fibre  stress  F. 


7       wLze 

wLze 
192000 

7      192000 

wLze 

wLze 

wLze 

c    24000^ 
7     24000F 

36000 
7     36000 

16800 
7      16800 

10800 
7      10800 

c         Lze 
I     24000F 

c  X    Lze 
7      192000 

c        7>2e 
7     36000 

7      16800 

c  X  L2e 
7      10800 

C'  f\                    -j-   n 

c         wLz 

c  ^    iyL2 

c  X  wL2 

c  X  w;L2 

c  X  w;L2 
wL3e 

II     24000F 

//      192000 

/7     36000 

//  v  16800 

.L/  —  AJ     x 
*  c         we 

,      wL3e 
~177.8E 

177.8EI 

\CX     ^e 

For  maximum 

wL3e 
2578100 

25781007 

\CX      ^6 

safe  deflection 
wL3e 

"  c        t^e 

L 
30' 

1422400 
14224007 

124460 
1244607 

80010 
800107 

L3e 
177.8EI 

L3e 
25781007 

14224007 

1244607 

L3e 
800107 

wL3 

wL3 

w;L3 

w;L3 

wL3 

sll77.8EI 

137.1A/  — 
'  iye 

For  directly  co 

-X0.0745L 
c 

maximum  safe  fi 
13.43c 

Actual  max 
wL*e 

me  -PI 

3/7 

3|    7 

~*      we      J 

For 

ec 

wL*e 

'  t^e                 '  iye 
mputing  7  from  —  . 

-X0.0253L      -X0.135 

bre  stress  and  deflection. 
39.50c               7.40c 

imum  deflection. 
wL*e                wL4e 

-X0.135L 
7.40c 

5333^7 

773285007 

426640007 

37331007 

23998507 

TABLES 

CASE  10A.    JOIST  WITH  ENDS  FIXED.    LOAD  UNIFORM 

TABLE   17 


Oak,  Wh.  Pine,  L.L.        Pine,  S.L.         Pine,  Wh. 

For  maximum  safe  fibre  stress  F. 


wL3e 


For  maximum  safe  deflection  — . 


wL3e 


wL3e 


Spruce 


7      wUe 
c     15600 

wL?e 
16800 

wL2e 
13200 

wUe 
10800 

wL2e 
13200 

*4> 

15600 

7 

16800 

7 

c 

13200 

/ 

c 

10800 

7 

c 

13200 

^2 

e-f. 

15600 

7 

c 

16800 
X  wL* 

7 

c 

13200 
X  wL* 

7 

c 

10800 
X  wV 

7 

c 

13200 
X  wL* 

1.12, 

3  A  wee 

130A/—           U 
>  wee 

*  wee 

1( 

^  wee 

i: 

*  wee 

133350 
1333507 

151130 
1511307 

106680 
1066807 

88900 
889007 

115570 

1155707 

L3e 

1155707 

1333507 

1511307 

L3e 
1066807 

889007 

wL* 

L-51.1-J/- 
*  t^e 

wL3 

53.3^ 
Vtw 

wL3 

47.4^- 
*  we 

wL3 

44.6^- 
*  iye 

w;L3 
48.7-^/1 

7=-X0.117L 


For  directly  computing  7  from  — . 

C 


-X0.113L   -X0.122L  -XO.H4L 


For  maximum  safe  fibre  stress  and  deflection. 
L=8.55c  8.98c  8.10c  8.27c 

Actual  maximum  deflection. 


wL4e 
45330507 


31998007 


26665007 


8.76c 


wL4e 
34664507 


66 


SIMPLIFIED  FORMULAS  AND  TABLES 


CASE  10s.    FLOORING  FIXED  AT  ENDS.    LOAD  UNIFORM 

TABLE   18 


General 


>-V 


wV- 


w  = 


400QFf2 
L2 


Hf 


w;: 


14.82# 
U.82EP 


3/14.82ff*3 
~"  \        ?« 


Steel  Cast  Iron         Fir,  Wash.       Hemlock 

For  maximum  fibre  stress  F 


52.8 

28001 
L2 

52.8* 


For  maximum  safe  deflection     r. 


21.8 


10374  £ 


21>8I 


For  maximum  safe  fibre  stress  and  deflection. 


42.4 

1800*2 
L2 


18.9 


6669^ 


18.9^ 


270^ 


Actual  maximum  deflection. 


A  = 


wL* 
444.4^^3 


3.70« 


wL* 


TABLES  67 

CASE  10s.    FLOORING  FIXED  AT  ENDS.    LOAD  UNIFORM 

TABLE  18 

Oak,  Wh.  Pine,  L.L.         Pine,  S.L.         Pine,  Wh.  Spruce 

For  maximum  safe  fibre  stress  F. 


v  ~~    nn   o 


For  maximum  safe  deflection  ^=. 


A  = 


50.9 

52.8 

46.9 

42.4 

46.9 

2600*2 

2800*2 

2200*2 

1800*2 

2200* 

L2 

L2 

L2 

L2 

L2 

50.9* 

52.8* 

46.9* 

42.4* 

46.9* 

22.3  23.3  20.7  19.5  21.3 

12597-j^  8892-^  7410  ^  9633-^ 


OO  Q/  OQ  Q/  OO  *7-t  1 Q  P\/  O1    Q/ 

ZZ.ot  Zo.ot  zu./c  ly.ot  zi.ot 


For  maximum  safe  fibre  stress  and  deflection. 
27*  4.50*  4.04*  4.12*  4.38* 

Actual  maximum  deflection. 
wL*  wL* 


333600*3  377740*3  266640*3  222200*3  288860*3 


68 


SIMPLIFIED  FORMULAS  AND  TABLES 


a 


o 


«  § 

g  a 


DQ 


CD 


2  g 

00    rH 


t~-    CO    O5    CO  00 

rH     OS     CO     CO  CO 

rH    CO    O>    OO  'rH 

rH  CO 


£  £ 

CO     »0 
T^    O5 


S5  £ 


CO   rJH   00 

rH    rH    I> 

^   CO   00 


3  8 


50 

rH    g 


§§    % 


00 


00  Tt< 

(N  TjH 

$*•  t*- 

rH  (M 


s 


O   l> 

00   <N 


IO    CO    Tjj    rH    1^    d    O5 

rH    CO    CO    rH    00 


o  ca 


rH    1O    <M    IO 


rH    CO    IO    l>    00 

rH    CO    00    CO    OO 


s 


00   CO   CO 
rH    TF    00 


CO   00   O   (N 


88  2 


8  S 


TABLES 


69 


o 


a 

^Q    -^H    -HH    CO    f^    CO 
rH     CM     Tfl     tO 

g  |  i  i  I 

I 

I 

iO   O   CM   Tt<   t^   OO 

rH     »O    CO    CO     CO    CM 

O    O5    00   t^   IO 

rH    CO     00    CO    t^ 

t>-    Oi   rH    rti    l>« 
rH    rH    rH 

CM 

rH 

R 

CO    CO   ^^    CO   CO    ^^ 

rH    IO    CM     rH     CO     GO 
rH     CM    CO    TJH 

CO    CO    ^D   CO    ^^ 
LO    IO    00    CO    rH 

CO   00   O   CO   CO 

r—  1    rH    rH 

1 

00 

rH 

CM   00   00   CM   O   CM 

rH    rH 

rH 

CO 

05    rH    ^    CO 

C^l   OO   CO   CO   O5 

»O   CO   00   O   C^ 

I 

3 

CO    00    rfri    CO    CO 

t^   OO   CO   CO    O2 
to   O5   >O   CO   (N 

Tfl    »O    l>    O2    rH 
rH 

1 

rH 

00    CM    <M    GO    O    00 
CO   t>-    C^    O    GO 

<N   (M   00   O   00 

O5    rH     ^H     ^^    CO 

CO    >O    CD    OO    O2 

CM 

to 

rH 

0 

rH 

C^   co   O   CO   ^ti 

rH     rH     C^ 

CO  .   Tf   iO   CO   00 

1 

00 

U^    rH    00    »0    CO    <M 
C^J    '^    00   CO    O5 

rH     rH     (M     CO    »0 
CO    ^*    CO    CO    Tt^ 
Cl    CO    ^    >O    CO 

1 

CO 

rH     CO    CO    O    ^ 

Ol   «0   (N   O   00 

rH    CM    CO    ^t^    "^ 

CO 

* 

rH     TjH    'CO    t^    CO 
CO    rH    CM    "^    CO    O5 

rH    rH     CO    1>    CO 
CO    t^    rH    CO    (M 
rH     rH     CM    CM    CO 

I 

CO 

<N   00   00   (M   O   <M 

00   00   (N   O   CM 
O5    CM    CO    O    TjH 

8 

« 

rH    IO    <M    rH    CO    00 
rH     (M    CO     TJH 

IO    tO    00    CO    rH 
CO   00   O   CO   CO 

rH    rH    rH 

1 

s 

rH    rH    C^    CO 

CO    O    00    00    rH 

iO   t>-   GO   O   CO 

r-(    rH 

co 
to 

rH 

rH 

rH    CO    CO    rH    |>    T* 

CO    CO    T^l    1>    rH 
CO    -^t1   to   CO   OO 

1 

fl  .   . 

51 

(N   -tf   CO   00   O   (N 

Tfi   CD   OO   O   CM 

* 

0)  £ 

70 


SIMPLIFIED  FORMULAS  AND  TABLES 


PROPERTIES  OF  CAST-IRON  LINTELS.     TABLE  21 


Sect. 

Dims. 

7 

c 

|"  Metal. 

7 

c 

1"  Metal. 

7 
c 

i"  Metal. 

I 

c 

7 

c 

7 

c 

L 

6  x6 
6x7 
6x8 

14.0 
15.7 
17.3 

24.2 
25.3 
26.3 

1.73 
1.61 
1.52 

15.8 
17.7 
19.6 

28.2 
29.5 
30.8 

1.78 
1.67 
1.57 

17.4 
19.6 
21.7 

31.9 
33.5 
34.9 

1.83 
1.71 
1.61 

6  xlO 

20.5 

27.9 

1.36 

23.2 

32.7 

1.41 

25.5 

37.1 

1.46 

L 

6  x!2 

7  x7 
7x8 

23.5 
22.9 
25.2 

29.2 
39.9 
41.2 

1.24 
1.74 
1.64 

26.5 

25.8 
28.5 

34.2 
46.5 
48.4 

1.29 
1.80 
1.70 

28.9 
28.7 
31.6 

38.8 
52.7 
54.9 

1.34 
1.84 
1.74 

8  x8 

35.2 

61.6 

1.75 

40.0 

72.3 

1.81 

44.4 

82.5 

1.86 

1 

8x10 
8x12 
9x8 

*    •   • 



.36.6 
41.7 
36.5 

74.6 

78.7 
97.1 

2.04 
1.89 
2.66 

40.6 
46.5 
40.9 

85.2 
89.7 
110.8 

2.10 
1.93 
2.71 

10  x!2 

.   .   . 

50.9 

130.1 

2.56 

56.9 

147.8 

2.60 

12  x!2 

74.4 

246.4 

3.28 

85.7 

284.2 

3.32 

6  x8 

20.8 

42.2 

2.03 

23.5 

48.9 

2.08 

26.0 

55.2 

2.12 

6  xlO 

24.4 

45.3 

1.87 

27.7 

53.1 

1.92 

30.6 

60.0 

1.96 

6x12 

28.0 

48.4 

1.73 

32.0 

56.9 

1.78 

35.0 

63.7 

1.82 

6  x  14 
6  x!6 

.    .    . 

35.4 
39.3 

59.1 
61.6 

1.67 
1.57 

39.2 
43.3 

67.0 
69.7 

1.71 
1.61 

6x18 

.... 

42.5 

63.3 

1.49 

47.1 

72.1 

1.53 

6  x20 

46.4 

65.4 

1.41 

50.8 

74.1 

1.46 

8x12 

49.5 

127.0 

2.57 

55.4 

144.6 

2.61 

8  x!4 

.  .  . 

55.4 

133.9 

2.42 

61.9 

152.3 

2.46 

8  x!6 

.... 

61.3 

139.7 

2.28 

68.3 

159.0 

2.33 

8  x!8 

66.8 

144.8 

2.17 

74.7 

165.0 

2.21 

8  x20 

72.6 

149.4 

2.06 

80.8 

170.4 

2.11 

8  x24 

92.9 

179.2 

1.93 

8  x28 

104.1 

186.5 

1.79 

10x20 

100.9 

280.4 

2.78 

114.8 

323.6 

2.82 

10  x24 

131.6 

342.0 

2.60 

10  x28 

147.6 

357.2 

2.42 

12  x20 

... 

130.7 

465.3 

3.56 

148.2 

533.7 

3.60 

12  x24 

170.1 

566.2 

3.33 

12  x28 

TABLES 


71 


PROPERTIES  OF  CAST-IRON  LINTELS.     TABLE  22 


Sect. 

Dims. 

I 

c 

1"  Metal. 

7 
c 

1J"  Metal. 

/ 

c 

1J"  Metal. 

I 

c 

/ 

c 

I 

c 

6  x6 

19.0 

35.4 

1.87 

21.4 

42.0 

1.96 

23.4 

47.7 

2.04 

6  x7 

21.3 

37.2 

1.75 

23.9 

44.2 

1.85 

26.3 

50.3 

1.93 

6  x8 

23.5 

38.7 

1.65 

26.2 

45.9 

1.75 

28.8 

52.6 

1.83 

6  xlO 

27.5 

41.2 

1.50 

30.9 

49.4 

1.60 

33.4 

56.0 

1.68 

L 

6  x!2 

7  x7 
7  x8 
8x8 

31.2 
31.1 
34.6 

48.5 

43.1 
58.7 
61.8 
92.2 

1.38 
1.89 
1.79 
1.90 

34.7 
35.4 
38.5 
43.4 

51.3 
70.1 

72.7 
106.6 

1.48 
1.98 
1.89 
2.46 

37.4 
38.9 
42.8 
45.4 

58.7 
80.6 
84.2 
115.6 

1.57 
2.07 
1.97 
2.55 

8x10 

44.3 

95.2 

2.15 

51.0 

114.2 

2.24 

56.7 

131.9 

2.33 

8  x!2 

51.0 

100.3 

1.97 

58.2 

120.3 

2.07 

65.4 

141.3 

2.16 

9  x8 

45.5 

124.3 

2.75 

52.6 

149.6 

2.85 

59.0 

172.7 

2.93 

«M« 

10x12 

63.3 

166.92.64 

84.0 

229.9 

2.74 

94.4 

267.2 

2.83 

12  x!2 

94.6 

318.5 

3.37 

111.6 

387.0 

3.47 

123.4 

437.7 

3.55 

6  X8 

28.3 

61.5 

2.17 

32.2 

72.7 

2.26 

35.2 

82.4 

2.34 

6  xlO 

33.3 

66.6 

2.00 

37.7 

78.9 

2.09 

41.4 

89.8 

2.17 

6  Xl2 
6  x!4 
6x16 

38.6 
42.5 
46.9 

71.0 
74.4 

77.4 

1.84 
1.75 
1.65 

42.8 
47.6 
52.1 

82.9 
87.1 
90.6 

1.94 
1.83 
1.74 

47.3 
52.5 
57.5 

95.5 
100.8 
105.2 

2.04 
1.92 
1.83 

6x18 

51.0 

80.0 

1.57 

56.9 

93.8 

1.65 

62.2 

108.9 

1.75 

6  x20 

54.9 

82.4 

1.50 

60.3 

95.3 

1.58 

66.8 

112.1 

1.68 

8  x!2 

60.8 

161.5 

2.66 

70.2 

192.9 

2.75 

78.5 

221.8 

2.83 

8x14 

68.1 

170.2 

2.50 

77.7 

203.8 

2.60 

87.4 

234.2 

2.68 

8  x!6 

75.1 

177.9 

2.37 

85.2 

209.4 

2.46 

96.6 

245.1 

2.54 

8  x!8 

82.0 

184.5 

2.25 

94.2 

221.3 

2.35 

104.5 

254.7 

2.43 

8  x20 

88.6 

190.4 

2.15 

102.0 

228.5 

2.24 

113.1 

263.6 

2.33 

8  x24 

101.3 

200.5 

1.98 

116.4 

240.8 

2.07 

128.7 

277.9 

2.16 

8  x28 

113.4 

208.7 

1.84 

129.9 

250.7 

1.93 

144.1 

289.6 

2.02 

10  x20 

125.5 

360.0 

2.87 

146.4 

428.6 

2.93 

165.3 

503.9 

3.05 

10  x24 

144.1 

380.1 

2.64 

166.3 

455.4 

2.74 

188.5 

533.6 

2.83 

10  x28 

153.5 

377.6 

2.46 

188.5 

480.6 

2.55 

211.3 

557.8 

2.64 

12  x20 

165.2 

600.8 

3.64 

194.3 

728.3 

3.75 

222.2 

849.9 

3.83 

12  x24 
12  x28 

189.1 
212.8 

637.1 
667.7 

3.37 
3.14 

223.0 
250.5 

773.7 
811.4 

3.47 
3.24 

254.2 

284.7 

902.5 
948.0 

3.55 
3.33 

72 


SIMPLIFIED  FORMULAS  AND  TABLES 


PROPERTIES  OF  CAST-IRON  LINTELS.     TABLE  23 


Sect. 

Dims. 

/ 
C 

|"  Metal. 

7 
c 

\"  Metal. 

7 
c 

\"  Metal. 

7 

c 

7 

c 

7 

c 

8  Xl6 

59.2 

155.5 

2.63 

68.4 

183.9 

2.69 

75.9 

207.7 

2.73 

8x18 

62.2 

162.3 

2.61 

74.6 

191.5 

2.57 

83.1 

216.8 

2.61 

8x20 

79.9 

196.3 

2.46 

89.8 

224.4 

2.50 

8  x24 

105.0 

238.3 

2.27 

8  X28 

10  x20 

111.9 

366.9 

3.28 

126.2:    420.4 

3.33 

10  x24 

146.8     447  7 

3.05 

10  x28 

12  x  20 

146.6 

607.0 

4.14 

166.0 

695.5 

4.19 

12  x24 

185.5 

729.4 

3.87 

12x28 

1"  Metal. 

\\"  Metal. 

1J"  Metal. 

8x16 

84.0 

232.4 

2.77 

97.1 

277.4  2.86 

107.8 

318.0 

2.95 

8x18 

91.5 

242.3 

2.65 

105.2 

289.4 

2.75 

117.5 

332.4 

2.83 

8x20 

98.6 

251.3 

2.55 

114.0 

300.7 

2.64 

137.5 

375.1 

2.73 

8x24 

112.7 

267.1 

2.37 

130.0 

319.8 

2.46 

144.5 

367.1 

2.54 

8x28 

126.1 

280.1 

2.22 

145.4 

335.7 

2.31 

160.7 

383.9 

2.39 

10  X20 

139.8 

471.3 

3.37 

163.6 

567.4 

3.47 

184.9 

655.9 

3.55 

10  x24 

161.6 

508.8 

3.15 

186.7 

604.8 

3.24 

210.5 

700.4 

3.33 

10  x28 

178.4 

527.8 

2.96 

209.2 

637.9 

3.05 

235.7 

737.6 

3.13 

12x20 

184.6 

782.5 

4.24 

218.8 

947.2 

4.33 

251.6 

1110.8 

4.42 

12  x24 

209.8 

834.7 

3.98 

248.2 

1011.0 

4.07 

283.5 

1117.8 

4.16 

12  x28 

234.9 

880.0 

3.75 

278.0 

1066.8 

3.84 

317.4 

1246.5 

3.93 

If"  Metal. 

12x28 

314.0 

1072.6 

3.42 

8  x28 

174.5 

434.2 

2.49 

10x28 

1  O  NX  Oft 

258.0 

OCA    Z, 

833.0 
1  /toQ  n 

3.23 
4  no 

14  X  Zo 

OOU.O 

14Uo.U 

.uz 

TABLE   OF  LOGAEITHMS 


74 


SIMPLIFIED  FORMULAS  AND  TABLES 
TABLE  OF  LOGARITHMS.    0  TO  499 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

0000 

0000 

3010 

4771 

6021 

6990 

7782 

8451 

9031 

9542 

1 

0000 

0414 

0792 

1139 

1461 

1761 

2041 

2304 

2553 

2788 

2 

3010 

3222 

3424 

3617 

3802 

3979 

4150 

4314 

4472 

4624 

3 

4771 

4914 

5051 

5185 

5315 

5441 

5563 

5682 

5798 

5911 

4 

6021 

6128 

6232 

6335 

6435 

6532 

6628 

6721 

6812 

6902 

5 

6990 

7076 

7160 

7243 

7324 

7404 

7482 

7559 

7634 

7709 

6 

7782 

7853 

7924 

7993 

8062 

8129 

8195 

8261 

8325 

8388 

7 

8451 

8513 

8573 

8633 

8692 

8751 

8808 

8865 

8921 

8976 

8 

9031 

9085 

9138 

9191 

9243 

9294 

9345 

9395 

9445 

9494 

9 

9542 

9590 

9638 

9685 

9731 

9777 

9823 

9868 

9912 

9956 

Diff. 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

41.5 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

37.9 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

34.9 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

32.3 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

30.1 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

28.1 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

26.4 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

25.0 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

23.5 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

22.3 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21.2 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

20.2 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

19.3 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

18.6 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

17.8 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

17.1 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

16.4 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

15.8 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

15.2 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

14.8 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

14.3 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

13.8 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

13.4 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

13.0 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

12.6 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

12.2 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

11.9 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

11.6 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

11.2 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

11.0 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

10  .  7 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

10.4 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

10.1 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

10.0 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

9.9 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

9.5 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

9.3 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

9.4 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

9.0 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

8.8 

TABLES 


75  / 


TABLE  OF  LOGARITHMS.     500  TO  999 


1 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

8.6 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

8.5 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

8.3 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

8.1 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

8.0 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

7.8 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

7.7 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

7.6 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

7.4 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

7.2 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

7.1 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

7.1 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

7.0 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

6.9 

64 

8062 

8069 

8075 

8082' 

8089 

8096 

8102 

8109 

8116 

8122 

6.8 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

6.7 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

6.6 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

6.5 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

63 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

6.2 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

6.1 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

6.0 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

6.0 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

5.9 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

5.8 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

5.7 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

5.6 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

5.5 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

5.5 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

5.4 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

5.3 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

5.3 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

5.3 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

5.2 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

5.1 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

5.1 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

5.0 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

4.9 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

4.8 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538. 

4.8 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

4.8 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

4.8 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

4.7 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

4.7 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

4.6 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

4.6 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

4.5 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

4.5 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

4.5 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

4.5 

76 


SIMPLIFIED  FORMULAS  AND  TABLES 


TABLE  OF  LOGARITHMS.     1000  TO  1499 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

100 

0000 

0004 

0009 

0013 

0017 

0022 

0026 

0030 

0035 

0039 

4.3 

101 

0043 

0048 

0052 

0056 

0060 

0065 

0069 

0073 

0077 

0082 

4.3 

102 

0086 

0090 

0095 

0099 

0103 

0107 

0111 

0116 

0120 

0124 

4.2 

103 

0128 

0133 

0137 

0141 

0145 

0149 

0154 

0158 

0162 

0166 

4.2 

104 

0170 

0175 

0179 

0183 

0187 

0191 

0195 

0199 

0204 

0208 

4.2 

105 

0212 

0216 

0220 

0224 

0228 

0233 

0237 

0241 

0245 

0249 

4.1 

106 

0253 

0257 

0261 

0265 

0269 

0273 

0278 

0282 

0286 

0290 

4.1 

107 

0294 

0298 

0302 

0306 

0310 

0314 

0318 

0322 

0326 

0330 

4.0 

108 

0334 

0338 

0342 

0346 

0350 

0354 

0358 

0362 

0366 

0370 

4.0 

109 

0374 

0378 

0382 

0386 

0390 

0394 

0398 

0402 

0406 

0410 

4.0 

110 

0414 

0418 

0422 

0426 

0430 

0434 

0438 

0441 

0445 

0449 

3.9 

111 

0453 

0457 

0461 

0465 

0469 

0473 

0477 

0481 

0484 

0488 

3.9 

112 

0492 

0496 

0500 

0504 

0508 

0512 

0515 

0519 

0523 

0527 

3.9 

113 

0531 

0535 

0538 

0542 

0546 

0550 

0554 

0558 

0561 

0565 

3.8 

114 

0569 

0573 

0577 

0580 

0584 

0588 

0592 

0596 

0599 

0603 

3.8 

115 

0607 

0611 

0615 

0618 

0622 

0626 

0630 

0633 

0637 

0641 

3.8 

116 

0645 

0648 

0652 

0656 

0660 

0663 

0667 

0671 

0674 

0678 

3.7 

117 

0682 

0686 

0689 

0693 

0697 

0700 

0704 

0708 

0711 

0715 

3.7 

118 

0719 

0722 

0726 

0730 

0734 

0737 

0741 

0745 

0748 

0752 

3.7 

119 

0755 

0759 

0763 

0766 

0770 

0774 

0777 

0781 

0785 

0788 

3.7 

120 

0792 

0795 

0799 

0803 

0806 

0810 

0813 

0817 

0821 

0824 

3.6 

121 

0828 

0831 

0835 

0839 

0842 

0846 

0849 

0853 

0856 

0860 

3.6 

122 

0864 

0867 

0871 

0874 

0878 

0881 

0885 

0888 

0892 

0896 

3.6 

123 

0899 

0903 

0906 

0910 

0913 

0917 

0920 

0924 

0927 

0931 

3.6 

124 

0934 

0938 

0941 

0945 

0948 

0952 

0955 

0959 

0962 

0966 

3.6 

125 

0969 

0973 

0976 

0980 

0983 

0986 

0990 

0993 

0997 

1000 

3.4 

126 

1004 

1007 

1011 

1014 

1017 

1021 

1024 

1028 

1031 

1035 

3.4 

127 

1038 

1041 

1045 

1048 

1052 

1055 

1059 

1062 

1065 

1069 

3.4 

128 

1072 

1075 

1079 

1082 

1086 

1089 

1092 

1096 

1099 

1103 

3.4 

129 

1106 

1109 

1113 

1116 

1119 

1123 

1126 

1129 

1133 

1136 

3.3 

130 

1139 

1143 

1146 

1149 

1153 

1156 

1159 

1163 

1166 

1169 

3.3 

131 

1173 

1176 

1179 

1183 

1186 

1189 

1193 

1196 

1199 

1202 

3.2 

132 

1206 

1209 

1212 

1216 

1219 

1222 

1225 

1229 

1232 

1235 

3.2 

133 

1239 

1242 

1245 

1248 

1252 

1255 

1258 

1261 

1265 

1268 

3.2 

134 

1271 

1274 

1278 

1281 

1284 

1287 

1290 

1294 

1297 

1300 

3.2 

135 

1303 

1307 

1310 

1313 

1316 

1319 

1323 

1326 

1329 

1332 

3.2 

136 

1335 

1339 

1342 

1345 

1348 

1351 

1355 

1358 

1361 

1364 

3.2 

137 

1367 

1370 

1374 

1377 

1380 

1383 

1386 

1389 

1392 

1396 

3.2 

138 

1399 

1402 

1405 

1408 

1411 

1414 

1418 

1421 

1424 

1427 

3.1 

139 

1430 

1433 

1436 

1440 

1443 

1446 

1449 

1452 

1455 

1458 

3.1 

140 

1461 

1464 

1467 

1471 

1474 

1477 

1480 

1483 

1486 

1489 

3.1 

141 

1492 

1495 

1498 

1501 

1504 

1508 

1511 

1514 

1517 

1520 

3.1 

142 

1523 

1526 

1529 

1532 

1535 

1538 

1541 

1544 

1547 

1550 

3.0 

143 

1553 

1556 

1559 

1562 

1565 

1569 

1572 

1575 

1578 

1581 

3.0 

144 

1584 

1587 

1590 

1593 

1596 

1599 

1602 

1605 

1608 

1611 

3.0 

145 

1614 

1617 

1620 

1623 

1626 

1629 

1632 

1635 

1638 

1641 

3.0 

146 

1644 

1647 

1649 

1652  1655 

1658 

1661 

1664 

1667 

1670 

2.9 

147 

1673 

1676 

1679 

1682  1685 

1688 

1691 

1694  1697 

1700 

2.9 

148 

1703 

1706 

1708 

1711 

1714 

1717 

1720 

1723  1726 

1729 

2.9 

149 

1732 

1735  |  1738 

1741 

1744 

1746  1749  1752  [  1755 

1758 

2.9 

TABLES 


77 


TABLE  OF  LOGARITHMS.     1500  TO  1999 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

150 

1761 

1764 

1767 

1770 

1772 

1775 

1778 

1781 

1784 

1787 

2.9 

151 

1790 

1793 

1796 

1798 

1801 

1804 

1807 

1810 

1813 

1816 

2.9 

152 

1818 

1821 

1824 

1827 

1830 

1833 

1836 

1838 

1841 

1844 

2.9 

153 

1847 

1850 

1853 

1855 

1858 

1861 

1864 

1867 

1870 

1872 

2.8 

154 

1875 

1878 

1881 

1884 

1886 

1889 

1892 

1895 

1898 

1901 

2.8 

155 

1903 

1906 

1909 

1912 

1915 

1917 

1920 

1923 

1926 

1928 

2.8 

156 

1931 

1934 

1937 

1940 

1942 

1945 

1948 

1951 

1953 

1956 

2.8 

157 

1959 

1962 

1965 

1967 

1970 

1973 

1976 

1978 

1981 

1984 

2.8 

158 

1987 

1989 

1992 

1995 

1998 

2000 

2003 

2006 

2009 

2011 

2.7 

159 

2014 

2017 

2019 

2022 

2025 

2028 

2030 

2033 

2036 

2038 

2.7 

160 

2041 

2044 

2047 

2049 

2052 

2055 

2057 

2060 

2063 

2066 

2.7 

161 

2068 

2071 

2074 

2076 

2079 

2082 

2084 

2087 

2090 

2092 

2.7 

162 

2095 

2098 

2101 

2103 

2106 

2109 

2111 

2114 

2117 

2119 

2.7 

163 

2122 

2125 

2127 

2130 

2133 

2135 

2138 

2140 

2143 

2146 

2.7 

164 

2148 

2151 

2154 

2156 

2159 

2162 

2164 

2167 

2170 

2172 

2.7 

165 

2175 

2177 

2180 

2183 

2185 

2188 

2191 

2193 

2196 

2198 

2.6 

166 

2201 

2204 

2206 

2209 

2212 

2214 

2217 

2219 

2222 

2225 

2.6 

167 

2227 

2230 

2232 

2235 

2238 

2240 

2243 

2245 

2248 

2251 

2.6 

168 

2253 

2256 

2258 

2261 

2263 

2266 

2269 

2271 

2274 

2276 

2.6 

169 

2279 

2281 

2284 

2287 

2289 

2292 

2294 

2297 

2299 

2302 

2.6 

170 

2304 

2307 

2310 

2312 

2315 

2317 

2320 

2322 

2325 

2327 

2.6 

171 

2330 

2333 

2335 

2338 

2340 

2343 

2345 

2348 

2350 

2353 

2.6 

172 

2355 

2358 

2360 

2363 

2365 

2368 

2370 

2373 

2375 

2378 

2.6 

173 

2380 

2383 

2385 

2388 

2390 

2393 

2395 

2398 

2400 

2403 

2.6 

174 

2405 

2408 

2410 

2413 

2415 

2418 

2420 

2423 

2425 

2428 

2.6 

175 

2430 

2433 

2435 

2438 

2440 

2443 

2445 

2448 

2450 

2453 

2.6 

176 

2455 

2458 

2460 

2463 

2465 

2467 

2470 

2472 

2475 

2477 

2.4 

177 

2480 

2482 

2485 

2487 

2490 

2492 

2494 

2497 

2499 

2502 

2.4 

178 

2504 

2507 

2509 

2512 

2514 

2516 

2519 

2521 

2524 

2526 

2.4 

179 

2529 

2531 

2533 

2536 

2538 

2541 

2543 

2545 

2548 

2550 

2.3 

180 

2553 

2555 

2558 

2560 

2562 

2565 

2567 

2570 

2572 

2574 

2.3 

181 

2577 

2579 

2582 

2584 

2586 

2589 

2591 

2594 

2596 

2598 

2.3 

182 

2601 

2603 

2605 

2608 

2610 

2613 

2615 

2617 

2620 

2622 

2.3 

183 

2625 

2627 

2629 

2632 

2634 

2636 

2639 

2641 

2643 

2646 

2.3 

184 

2648 

2651 

2653 

2655 

2658 

2660 

2662 

2665 

2667 

2669 

2.3 

185 

2672 

2674 

2676 

2679 

2681 

2683 

2686 

2688 

2690 

2693 

2.3 

186 

2695 

2697 

2700 

2702 

2704 

2707 

2709 

2711 

2714 

2716 

2.3 

187 

2718 

2721 

2723 

2725 

2728 

2730 

2732 

2735 

2737 

2739 

2.3 

188 

2742 

2744 

2746 

2749 

2751 

2753 

2755 

2758 

2760 

2762 

2.2 

189 

2765 

2767 

2769 

2772 

2774 

2776 

2778 

2781 

2783 

2785 

2.2 

190 

2788 

2790 

2792 

2794 

2797 

2799 

2801 

2804 

2806 

2808 

2.2 

191 

2810 

2813 

2815 

2817 

2819 

2822 

2824 

2826 

2828 

2831 

2.2 

192 

2833 

2835 

2838 

2840 

2842 

2844 

2847 

2849 

2851 

2853 

2.2 

193 

2856 

2858 

2860 

2862 

2865 

2867 

2869 

2871 

2874 

2876 

2.2 

194 

2878 

2880 

2882 

2885 

2887 

2889 

2891 

2894 

2896 

2898 

2.2 

195 

2900 

2903 

2905 

2907 

2909 

2911 

2914 

2916 

2918 

2920 

2.2 

196 

2923 

2925 

2927 

2929 

2931 

2934 

2936 

2938 

2940 

2942 

2.1 

197 

2945 

2947 

2949 

2951 

2953 

2956 

2958 

2960 

2962 

2964 

2.1 

198 

2967 

2969 

2971 

2973 

2975 

2978 

2980 

2982 

2984 

2986 

2.1 

199 

2989 

2991 

2993 

2995 

2997 

2999  3002 

3004 

3006 

3Q08 

2.1 

/.To 


i^IL^^^Perwi. '        VUctttoa  is  mad|  g°|oj° 

1 


SEP  16 


10m-4,'23 


308540 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


